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f^'  This  Question-Book,  not  being  an  exclusive  text-book,  but  a 
means  of  additional  illustration,  does  not  necessarily  require  to  be 
formally  adopted  by  Boards  of  Education,  but  can  be  introduced  and 
used  at  the  option  of  Teachers. 

For  hints  on  the  object  and  proper  use  of  these  Questions,  Teachers 
are  referred  to  the  *'  Preface,"  and  "Advantages,"  see  pages  2,  3, 4. 

STANDARD    SERIES     OF    ARITHMETICS: 
By   JAMES    S.    EATON, 

Principal  of  the  English  Department  of  Thillips  Academy,  Andover,  Mass. 
I.    Primary  Arithmetic,  100  pp.    Beautifully  illustrated. 
II.    Intellectual  Arithmetic,  176  pp.    On  a  progressive  plan. 

III.  Common  School  Arithmetic,  312  pp.  A  complete  practical  text-book. 

IV.  High  School  Arithmetic,  356  pp.    A  thorough  and  exhaustive 

treatise. 

ALSO 

Eaton's  Grammar  School  Arithmetic,  336  pp.  This  is  the  **  Common 
School "  with  24  additional  pages,taken  from  the  "  High  School.**  Designed 
for  schools  so  graded  as  to  require  but  one  book  on  Written  Arithmetic. 

Eaton's  Elements  of  Written  Arithmetic,  about  200  pp.  In  Press. 
Designed  for  Elementary  Classes. 

This  is  the  best  consecutive  series  by  the  same  author,  on  the  latest  im- 
proved plan,  and  adapted  to  the  wants  of  Primary,  Intermediate,  Grammar, 
and  High  Schools,  Academies,  and  Normal  Schools.  The  popularity  of 
Eaton's  Books,  and  the  largely  increasing  demand,  may  well  sustain  their 
claim  to  be  called  a 

NEW  NATIONAL   STANDARD  SERIES. 

***  Teachers  experiencing  the  disadvantages  of  using  text-books  by  dif- 
ferent authors,  or  old,  revamped  and  re-revised  books,  will  avoid  this  use  of  a 
multiplicity  of  different  editions  by  the  same  author,  constantly  appearing, 
by  adopting  the  clear,  scientific,  concise,  and  practical  system  prepared  by 
Mr.  Eaton.    These  Arithmetics  are 

Exclusively  used  in  the  Public  Schools  of  Boston, 
and  very  extensively  in   Massachusetts  and   New  England.    They  are  the 
authorized  and  exclusive  text-books  in  the  State  of  California.    They 
are  also  used  in  many  hundred  cities  and  towns,  where  the  best  schools  are 
maintained,  in  all  parts  of  the  United  States. 

Very  low  terms  are  offered  for  their  first  introduction,  or  they  may  be  in- 
troduced gradually,  at  the  regular  prices,  as  new  classes  are  formed. 

Send  for  Descriptive  Catalogue.  Specimen  copies  of  the  Arithmetics  fur- 
nished for  examination,  with  reference  to  introduction,  on  receipt  of 
Postage.  Primary  5  cents,  Intellectual  10  cents.  Common  School  and  High 
School  20  cents  each.    Address, 

TAGGARD  &  THOMPSON,  29  Cornhill,  Boston. 


QUESTIONS 


PEINCIPLES  OF  AEITHMETIC, 


TO  INDICATE  AN  OUTLINE  OF  STUDY; 

TO  INCITE  AMONG  PUPILS 

A  SPIRIT  OP  INDEPENDENT  INQUIRY; 

AND   ESPECIALLY  FITTED   TO   FACILITATE 

A  THOROUGH  SYSTEM  OF  REVIEWS. 

ADAPTED  TO  ANT  TEXT-BOOKS  AND  TO  ALL  GRADES  OP  LEAENEfiS. 

By  JAMES    S.   EATON,  M.   A. 

AUTHOR  OP  A  SERIES  OP  ARITHMETICS,  ETC. 


*'It  should  be  the  chief  aim,  in  teacliing'  Arithmetic,  to  lead  the  learner  to  a  clear  understand- 
ing of  the  Peinciples  of  the  Science."  — IIox.  Johx  D.  Puilbkick,  Sup't  Boston  Schools. 


BOSTON: 

TAGGAE.D    AND    THOMPSON. 

18  66. 


TO    TEACHERS. 

THE    ADVANTAGES    OF    USING    THESE    QUESTIONS. 

1.  They  are  separate  from  any  text-book,  and  equally  well 
adapted  to  all  text-books.  On  this  account  they  present  all  the 
benefits  of  the  Question  Method^  and  none  of  its  defects. 

2.  They  indicate  a  definite  outline  of  study,  and  thus  afford  a 
substantial  and  useful  guide  to  the  pupil  in  the  preparation  of  his 
lesson. 

3.  They  incite  the  pupil  to  inquiry,  awakening  that  thirst  for 
knowledge  which  is  the  best  motive  to  its  acquirement. 

4.  They  open  up  the  several  subjects  by  such  short  and  sugges- 
tive steps,  one  question  following  upon  another  in  the  chain,  that  the 
pupil  is  thus  led  to  follow  out  and  develop  the  subject  for  himself. 

5.  By  inciting  the  pupil  to  inquiry,  and  by  guiding  him  in  de- 
veloping the  subject  for  himself,  they  subserve  the  highest  and  only 
true  style  of  teaching,  namely,  to  draw  out  and  develop  iJie  facul- 
ties, and  thus  lead  the  pupil,  instead  of  dictating  to  him  and  driv- 
ing him. 

6.  They  afford  the  best  means  for  frequent  reviews  and  exam- 
inations, since  it  is  the  Principles  of  Arithmetic  that  should  be 
reviewed,  and  not  the  mechanical  operations.  ' 

7.  The  use  of  these  Questions  will  not  fail  to  ground  the  prin- 
ciples of  Arithmetic  in  the  mind  of  the  pupil,  and  thus  give  him 
the  Key  which  will  command  all  practical  operations. 


Entered,  according  to  act  of  Congress,  in  the  year  1805, 

BY  TAGGARD  AND  THOMPSON, 

In  the  Clerk's  Oflace  of  the  District  Court  of  the  District  of  Massachusettsu 


PEEFACE. 


These  Questions  are  offered  to  the  intelligent  Teachers  of  this 
country  with  the  hope  that  they  may  serve  as  an  essential  help  in  teach- 
ing the  important  branch  of  Arithmetic.  The  plan  of  this  book  is  in 
some  respects  new,-  and  it  is  thought  that  the  use  of  it  will  tend  to 
promote  a  more  thorough  and  successful  method  than  would  perhaps 
otherwise  be  attained.  It  is  generally  agreed  that  the  subject  of 
Arithmetic  is  apt  to  be  taught  too  mechanically,  too  much  by  mere 
rote,  —  by  **  ciphering  "  from  formulas  rather  than  by  an  intelligent 
discussion  of  principles.  Instead  of  directing  the  pupil  **to  cipher 
according  to  rule,"  he  should  rather  be  taught  to  perform  examples 
by  analysis,  according  to  principles  which  he  has  mastered,  and 
whose  wide  application  he  has  been  led  fully  to  understand. 

If  one  be  well  grounded  in  the  principles  of  a  science,  he  has  con- 
stantly at  his  command  the  Key  to  all  operations  pertaining  to  it. 
He  is  then  Master  of  the  Situation.  So  in  Arithmetic,  it  is 
more  important  that  the  pupil  should  know,  and  be  able  to  tell,  upon 
what  principle  any  given  operation  depends,  than  that  he  should  be 
able  to  solve,  according  to  a  set  formula,  which  he  does  not  under- 
stand, any  number  of"  similar  examples  whose  answers  are  all  before 
hi?n.  That  he  should  spend  montlis  and  years  of  his  schooling-  m 
adding,  multiplying,  and  dividing,  simplij  as  an  exercise,  is  no  less 
absurd  than  that"  a  mechanic  should  exercise  with  dumb-bells  and 
clubs  to  perfect  the  muscles  of  his  arm,  before  he  shall  -touch  the 
blacksmith^s  hammer  or  the  carpenter's  chisel.  Those  few  who  advocate 
so  much  practice  with  abstract  numbers  in  the  fundamental  rules,  in 
order  to  perfect  the  pupil  in  addition,  &c.,  before  proceeding  further 


4  PREFACE. 

(since  no  subsequent  page  of  the  Arithmetic  is  without  practice  in 
one  or  all  of  the  fundamental  rules),  remind  one  of  the  person  who 
advised  the  news-carrier,  whose  business  required  him  to  use  his  legs 
all  'day,  to  take  a  walk  of  a  few  miles  for  exercise  before  beginning 
his  day^s  work. 

Practical  examples  sliould  illustrate  priy^ciples,  but  should  not  hurij 
tliem  out  of  sight*  Facility  in  combining  numbers  with  celerity  and 
accuracy  attends  only  upon  long-continued  and  uninterrupted  prac- 
tice. When  the  occasion  for  such  practice  comes,  as  with  the  ac- 
countant and  book-keeper,  this  facility  also  comes. 

The  use  of  these  Questions  will,  it  is  believed,  awaken  curiosity 
and  stimulate  a  spirit  of  inquiry  among  pupils,  and  thus  interest  them 
in  the  subject,  a  thing  of  paramount  importance. 

The  only  way  in  which  the  principles  of  any  science  can  be  well 
and  thoroughly  established  in  the  mind,  is  by  constant  reviewing^  and 
to  this  end,  it  is  believed  that  no  method  can  surpass  the  use  of  theso 
Questions. 

They  are  adapted  to  any  text-book,  and  each  scholar  being  pro- 
vided with  a  copy  of  the  Question-book,  can  seek  wherever  his  choice 
may  dictate  for  the  best,  and  most  appropriate  answers. 

It  is  proper  to  state  that  though  these  Questions  are  sufficiently 
based  upon  material  of  Mr.  Eaton's  to  entitle  him  to  be  called  the 
author,  the  sudden  decease  of  this  distinguished  teacher  has  devolved 
upon  another  the  labor  of  preparing  them  for  the  press. 

The  Editor  takes  pleasure  in  acknowledging  his  obligations  to  the 
several  eminent  teachers  who  examined  the  proof-sheets  of  these 
Questions,  to  whom  he  is  indebted  for  valuable  suggestions. 

It  is  hoped  that  the  generous  favor  so  universally  shown  towards 
Mr.  Eaton's  Series  of  Arithmetics  will  be  extended  to  the  **  Ques- 
tions," and  that  this  endeavor  to  aid  his  fellow-teachers  will  meet 
with  their  cordial  approbation. 

The  Editor. 

Boston,  INfovEMBER  20,  1865. 


QUESTIO^^S. 


SECTION  I. 

NOTATION  AND   NmyiERATION. 

1.  What  is  Arithmetic  ? 

2.  Define  the  word  science* 

3.  Define  the  word  aH^ 

4.  What  is  a  number  ?     Give  an  example. 
6.  What  is  a  unit  ?     Give  an  example. 

6.  What  is  a  concrete  number?    Name  one. 

7.  What  is  an  abstract  number?     Name  one. 

8.  How  many  fundamental  operations  in  Arithmetic  ? 
Define  the  word  fundamental. 
What  is  Notation  ?    ^ 
What  is  Numeration  ? 

What  is  the  exact  meaning  of  the  word  notation  ? 
How  many  methods  of  notation  are  employed  ? 
Which  is  most  convenient  in  Arithmetic  ? 

15.  How  many  and  what  are  the  figures  of  the  Arabic  notation  ? 

16.  Whence  the  name  of  this  method  ? 
AVhat  are  the  Arabic  figures  sometiines  called  ? 
Whence  the  name  digit  7 
What  is  a  significant  figure  ? 

What  is  the  largest  number  a  single  figure  can  express  ? 
JHow  is  any  number  from  ten  to  one  hundred  expressed? 

22.   How  is  a  larger  number  expressed  ? 
t23.  What  is  understood  by  the  place  a  figure  occupies  ? 
24.  Where  is  the  &rst placCy  and  what  is  it  called? 
\* 


6  ARITHINIETIC. 

25.  The  second  place,  and  what  called?  third?  fourth?  &c. 

26.  What  Is  the  simple  value  of  a  figure  ? 

27.  What  is  the  local  value  of  a  figure  ? 

28.  What  is  the  effect  of  removing  a  figure  one  place  to  the  left? 

29.  Removing  It  two  places  to  the  left?    Three?  four?  &c. 

30.  What  principle  is  thus  made  evident? 

31.  Define  the  word  principle, 

32.  What  is  the  use  of  the  zero  (0)  ? 

33.  How  many  and  what  are  the  methods  of  numerating? 

34.  How  many  figures  comprise  a  period  in  each  method  ? 

35.  Which  is  mostly  used  in  this  country  ? 

'36.   Name  the  first  six  periods  in  the  French  method." 

37.  What  is  the  first  step  In  numerating  ? 

38.  What  is  the  second  step  ? 

39.  Explain  how  any  given  number  may  be  written. 

40.  Explain  the  process  of  reading  any  number. 

41.  Illustrate  the  two  preceding  questions  by  examples. 

42.  How  does  the  English  method  differ  from  the  French? 

43.  How  many  and  what  characters  are  employed  In  Roman  notation  ? 

44.  Upon  how  many  principles  are  the  letters  combined  in  Roman 
notation? 

45.  Name  and  Illustrate  each  of  them. 

46.  For  what  is  Roman  notation  chiefly  used  ? 


SECTION  n. 


1.  What  Is  Addition  ?     Give  an  example. 

2.  What  is  meant  by  the  sum  or  amounts 

3.  What  Is  the  sign  of  addition,  and  how  used? 

4.  What  is  the  sign  Indicating  dollars  ? 

5.  What  is  the  sign  of  equality,  and  how  used? 

6.  How  are  numbers  written  for  addition  ? 

7.  Which  column  is  first  added  ?  next  added  ? 

8.  Where  Is  the  sum  of  each  column  placed  ? 

9.  What  is  done  if  the  amount  of  any  column  exceeds  ten^ 

10.  Upon  what  principle  does  this  last  process  depend? 

11.  Explain  the  process  of  addition  by  an  example. 


ARITHMETIC. 


12.  Must  numbers  to  be  added  be  of  the  same  kind  ?     Why  ? 

13.  What  are  the  methods  oi proof  in  addition?     Illustrate; 

14.  Define  the  word  proof. 


SECTio:Nr  III. 

SUBTrwlCTION. 

1.  What  is  Subtraction  ?     Give  an  example. 

2.  AYhat  is  the  greater  number  called  ? 

3.  What  is  the  less  number  called  ? 

4.  What  is  the  result  obtained  by  this  process  called  ? 

5.  What  is  the  sign  of  subtraction,  and  how  used? 

6.  How  are  numbers  written  for  subtraction  ? 

7.  With  which  figure  do  we  begin  to  subtract  ? 

8.  Where  is  the  difference  or  remainder  written  ? 

9.  If  any  figure  in  the  upper  number  is  less  than  the  one  under- 
neath it,  what  is  to  be  done  ? 

10.  How  then  do  we  proceed  with  the  next  column  ? 

11.  Upon  what  principle  does  this  operation  depend? 

12.  Illustrate  the  principle  last  named. 

13.  Describe  the  complete  process  of  eubtraction,  and  give   an 
example. 

14.  What  is  the  method  oi  proof  ^    The  reason  for  it. 

15.  If  the  difference  be  added  to  the  subtrahend^  what  is  ob- 
tained ? 

16.  If  the  difference  be  subtracted  from  the  minuend,  what  is  ob- 
tained? 

11,   Of  what  operation  is  Subtraction  the  opposite  ? 


SECTioN^  rv. 

MULTIPLICATION. 

1.  What  is  Multiplication  ?     Give  an  example. 

2.  W^hat  process  does  it  resemble,  and  how  does  it  differ  from  it  ? 

3.  What  is  the  name  of  the  number  to  be  multiplied  ? 

4.  The  name  of  the  number  which  we  multiply  by  ? 


8  ARITHMETIC. 

5.  Wliat  is  the  result  obtained  by  the  process  called  ?  « 

6.  What  are  the  multiplier  and   multiplicand,    taken  together, 
called  ? 

7.  Define  the  word  factor, 

8.  AVhat  is  the  sign  of  multiplication,  and  how  used  ? 

9.  How  are  the  numbers  usually  written  in  multiplication? 

10.  Can  the  multiplier  be  a  concrete  number  ? 

11.  What  is  the  first  step  towards  multiplying  by  a  single  figure? 

12.  AVhere  are  the  units  of  the  product  written? 

13.  What  is  done  with  the  tens,  if  any  are  obtained? 

14.  Upon  what  principle  does  this  "  carrying"  depend? 

15.  Describe  the  full  process    when  the  multiplier  consists  of  a 
single  figure, 

16.  Illustrate  the  above  process  by  an  example. 

17.  In. multiplying  by  more  than  one  figure,  where  is  the  first  figure 
in  each  partial  product  written  ? 

18.  Give  the  reason  for  so  writing  it.  . 

19.  What  is  meant  by  **  partial  product?  " 

20.  How  is  the  complete  product  obtained? 

21.  Explain  the  full  process  of  multipljing  by  any  number  of  fig- 
ures? 

22.  Illustrate  the  above  process  by  an  example. 

23.  May  the  multiplier  and  the  multiplicand  exchange  pbces  ? 

24.  What  is  the  method  of  proof  in  multiplication  ? 

25.  What  is  a  composite  number  ?    ' 

26.  What  are  factors  ? 

27.  How  may  you  multiply  by  a  composite  number  ? 

28.  Illustrate  the  above  process,  and  give  the  reason  for  it. 

29.  How  may  a  number  be  multiplied  by  a  unit  with  any  number  of 
ciphers  at  its  right  ? 

30.  How,  when  there  are  ciphers  at  the  right  of  both  multiplier  and 
multiplicand? 

31.  How,  when  there  are  ciphers  between  the  significant  figures  of 
the  multiplier  ? 

.     32.    Illustrate  and  give  the  reason  for  the  last  three  processes. 

33.  Name  any  other  methods  of  contraction  in  multiplication  which 
you  know. 

34.  Define  the  word  contraction. 


ARITHMETIC. 


SECTION  V. 


DIVISION. 


1.  What  is  Division?     Give  an  example. 

2.  What  process  does  it  resemble,  and  how  does  it  differ  from  it  ? 

3.  What  is  the  number  to  be  divided  called  ? 

4.  What  is  the  number  by  which  we  divide  called  ? 

5.  AVhat  is  the  result  called  ? 

6.  AYhen  the  number  cannot  be  exactly  divided,  what  is  that  part 
of  the  dividend  left  called  ? 

7.  What  is  the  most  common  sign  of  division,  and  its  use  ? 

8.  What  other  signs  of  division  are  there  ? 

9.  Write  all  the  signs  of  division,  and  tell  their  use. 

10.  How  many  ways  of  performing  division,  and  what  are  they 
usually  called  ? 

11.  How  are  the  numbers  usually  written  for  division  ? 

12.  What  is  the  first  step  when  the  divisor  consists  of  but  one  fig- 
ure ? 

13.  What  is  the  next  step,  and  so  on  ? 

14.  Of  what  order  is  any  quotient  figure  ? 

15.  Describe  the  complete  process  of  Short  Division  ?    ** 

16.  Illustrate  the  above  process  by  performing  an  example. 

17.  When  is  the  division  said  to  be  complete  ? 

18.  When,  is   the  dividend  said  to  be   indivisible  by  a  number? 

19.  AVhat  is. an  exact  divisor? 

20.  When  the  divisor  is  so  large  as  to  require  all  the  process  to 
be  written  out,  what  kind  of  .divlson  is  it  ? 

21.  What  Is  the  first  step  In  the  process  of  Long  Division  ?  The 
second?     Tli«  third?     The  fourth? 

22.  When  the  remainder  (In  partial  division)  and  the  next  figure 
of  the  dividend  brought  down,  will  not  contain  the  divisor,  what  is 
to  be  done? 

23.  If  the  product  of  the  divisor  into  the  quotient  figure  exceeds 
any  partial  dividend,  what  Is  to  be  done  ? 

24.  What  if  the  last  remainder  equals  or  exceeds  the  divisor? 

25.  Of  what  is  division  the  opposite  ? 

26.  How,  then,  would  you  prove  division  ? 

27.  Upon  what  principle  does  this  proof  depend  ? 


10  AEITHMETIC. 

28.  How  is  division  by  a  composite  numbf^r  porformed?* 

29.  Illustrate  the  above  process  by  an  example. 

30.  If  there  arc   several  remainders  how  is  the  true  remainder 
found  ? 

31.  Explain  the  process   of  (inding  the  true  remainder,  by  an 
example. 

32-.    Give  the  reasons  for  the  steps  taken  in  the  last  process  ? 

33.  How  do  you  divide  by  a  unit  with  any  number  of  ciphers  at 
its  right  ? 

34.  Upon  what  principle  does  this  give  the  correct  result? 

35.  What  is  the  reason  for  this  process  ? 

36.  How  may  we  proceed  when  there  is  one  or  more  ciphers  at  the 
right  of  the  divisor  ? 

37.  How  is  the  true  remainder  found  in  this  case  ? 

38.  Can  you  name  any  other  contractions  in  division.^ 

39.  The  divisor  and  quotient  are  factors  of  what? 

40.  How  is  the  dividend  found  from  the  divisor  and  quotient? 

41.  How  is  the  divisor  found  from  the  dividend  and  quotient? 


SECTION  VI. 

GENERAL    PRINCIPLES    OF    DIVISION. 

1.  Will  a  large  dividend  and  large  divisor  always  produce  a  large 
quotient? 

2.  Will  a  small  dividend  and  a  small  divisor  always  produce  a 
small  quotient? 

3.  Does,  then,  the  size  of  the  quotient  in  division  depend  upon 
the  absolute  size  of  the  divisor  and  dividend  ?  ^ 

4.  Define  the  word  absolute.  "* 

5.  Define  the  word  relative. 

6.  Upon  what  does  the  size  or  value  of  the  quotient  depend  ? 

7.  If  the  divisor  remains  unaltered,  how  does  multiplying  the 
dividend  affect  the  quotient?     Give  an  example.* 

8.  Tf'tlic  divibor  remains  unaltered,  how  does  dividing  the  divi- 
dend affect  the  quotient  ?   ~  Give  an  example. 

*  In  the  illustrative  examples,  let  tlie  dividend  be  written  above  tlie  line,  and 
the  divisor  below  the  line,  as  ]  2 

4 


ARITHMETIC.  11 

9.   If  the  dividend  remains  unaltered,  how  does  multiplying  the 
divisor  affect  the  quotient?     Give  an  example. 

10.  If  the  dividend  remains  unaltered,  how  does  dividing  the  divi- 
sor affect  the  quotient?     Give  an  example. 

11.  How  does  increasing  or  decreasing  the  dividend,  in  either  case, 
affect  the  quotient  ?  * 

12.  How  does  increasing  or  decreasing  the  divisor,  in  either  case, 
affect  the  quotient  ? 

13.  Multiplying  or  dividing  both  dividend  and  divisor  by  the  same 
number,  how  in  either  case  affects  the  quotient  ? 

14.  AYhat  three  general  principles,  in  regard  to  the  quotient,   may 
be  here  laid  down  ? 

15.  Give  examples  illustrating  each. 


SECTIO:^T  YII. 

REDUCTION. 

1.  What  is  a  Compound  Number?     Give  an  example. 

2.  What  is  a  Simple  Kumber?     Give  an  example. 

3.  Are  abstract  numbers  simple  or  compound? 

4.  What  is  a  Denominate  Number  ?        _ 

5.  Is  the  expression  3  weeks,  2  pints,  and  6  yards,   a  compound 
number? 

6.  Give  a  compound  number. 

7.  How  does  it  differ  from  the  numbers  in  question  5  ? 

8.  Is  dvery  compound  number  a  denominate  number  ? 

9.  Is  every  denominate  number  a  compound  number  ? 

10.  What  is  Reduction  ? 

11.  How  many  kinds  of  Reduction,  and  what  are  they? 

12.  Why  are  they  so  named  ? 

13.  Which  of  the  fundamental  operations  is  employed  in  Reduction 
Descending? 

14.  Give  an  example  of  the  above. 

15.  Explain  the  full  process   of  Reduction  Descending,   by   an 
example. 

I    16.  Which  of  the  fundamental  operations  is  employed  in  Reduction 

|A.scending_? 

^    17.   Give  an  example. 


12  ARITHMETIC. 

18.  Explain  tlie  full   process    of  Reduction  Ascending,  by  an 
example. 

19.  How  Is  each  of  these  processes  proved? 

20.  What    is    English    Money?      Repeat  the  table.     Give  the 
abbreviations  indicating  the  denominations. 

21.  What  is  meant  by  a  scale  in  Reduction  ? 

22.  Give  an  example  of  an  ascending  and  descending  scale. 

23.  What  is  the  scale  for  English  Money  ^ 

24.  For  what  is  Troy  Weight  used  ? 

25.  Repeat  the  table  for  Troy  Weight.     Give  the  abbreviations 
which  mark  the  denominations. 

26.  Give  its  descending  scale.     Its  ascending  scale. 

27.  For  what    is  Apothecaries'  Weight  used?     Give   the  table. 
Give  the  signs  which  mark  the  denominations. 

28.  By  what  weight  aj-e  medicines  bought  and  sold  ? 

29.  What  denomination  in  the  above  weights  are  alike  ? 

30.  For  what  is  Avoirdupois  Weight  used? 

31.  Repeat  the  table.      Give   the  abbreviations  which  mark  the 
denominations.     Give  the  scale. 

32.  How  many  pounds  now  make  a  ton  ?  How  many  formerly  ? 

33.  What  is  a  long  or  gross  ton  ?     A  short  ton  ? 

34.  Where  is  the  gross  ton  now  used  ? 

35.  One  pound  Avoirdupois  equals  how    many  grains  of  Troy  or 
Apothecaries'  Weight? 

36.  How  many  grains  heavier  is  the  Avoirdupois  pound  than  the 
Troy? 

37.  In  exchanging  a  quantity  of  gold  dust  for  cotton,  by  what 
weight  would  each  be  weighed  ? 

38.  For  what  is  Cloth  Measure  used  ? 

39.  Repeat  the  table  of  Cloth  Measure.     Give  the  abbreviations 
which  mark  the  denominations. 

40.  For  what  is  Long  Measure  used? 

41.  Repeat  the  table  of  Long  Measure.      Give  the  abbreviations 
which  mark  its  denominations. 

42.  Give  the  scale  of  Long  Measure. 

43.  How  long  is  a  degree  upon  a  circle  of  the  earth  ? 

44.  For  what  is  Chain  Measure  used  ? 

45.  Repeat  the  table  of  Chain  Measure.     Give  the  abbreviations 
which  mark  its  denominations. 

46.  What  is  the   difference  between  Long  Measure  and  Chain 
Measure  ? 


^     ARITHMETIC.        •  13 

47.  For  what  is  Square  Measure  used? 

48.  Repeat  the  table  of  Square  Measure.  Give  the  abbreviations, 
of  the  denominations. 

49.  Give  the  scale  of  Square  Measure. 

50.  Define  the  word  area, 

51.  Define  the  word  angle, 

52.  What  is  a  rectangle  ? 

53.  What  is  each  angle  of  a  rectangle  called  ? 

54.  How  is  the  area  of  a  rectangle  obtained  ? 

55.  How  is  the  breadth  of  a  rectangle  found,  when  the  area  and 
length  are  given  ? 

56.  How  is  the  length  found,  when  the  area  and  width  are  given  ? 
57:   What  is  Solid  or  Cubic  Measure  ? 

58.  Repeat  the  table.     Give  the  abbreviations. 

59.  What  is  a  prism  ?     A  rectangular  prism  ? 

60.  Wiiat  is  a  cube  ?  • 

61.  How  are  the  solid  contents  of  a  rectangiilar  prism,  or  a  cube 
found? 

62.  How  is  the  depth,  length,  or  breadth  of  a  rectangular  prism 
or  cube  found,  if  the  solid  contents  and  the  area  of  one  face  is 
known? 

63.  For  what  is  Liquid  Measure  used  ?  ^ 

64.  Repeat  the  table.     Give  the  abbreviations. 
Qb,   What  is  the  standard  unit  of  Liquid  Measure  ? 
Qi^.    How  majny  cubic  inches  does  this  unit  contain  ? 

67.  How  many  cubic  inches  does  a  gallon  contain? 

68.  What  are  the  different  names  for  casks  containing  from 
50  to  150  gallons  or  more  ? 

69.  For  what  is  Dry  Measure  used?  ^ 

70.  Repeat  the  table  of  Dry  Measure.  Give  the  abbreviations 
of  its  denominations. 

71.  How  many  solid  inches  does  a  bushel  contain? 

72.  How  many  gallons  does  a  bushel  contain  ? 

73.  What  are  the  dimensions  of  the  common  bushel  measure? 

74.  For  what  is  Time  used  ? 

75.  What  are  the  natural  divisions  of  Time  ? 

76.  What  are  the  artificial  divisions  of  Time? 

77.  Repeat  the  table  for  Time  Measure.     Give  the  abbreviations. 

78.  Give  the  scale  of  Time  Measure  ? 

0 

79.  What  are  the  names  of  the  calendar  months  P 


14  ARITHMETIC. 

80.  How  many  days  in  each  ? 

81.  "What  is  meant  by  a  solar  year,  and  what  is  its  length? 

82.  What  is  meant  by  a  lunar  month  ? 

83.  What  is  meant  by  a  leap  year? 

84.  For  what  is  Circular  Measure  used  ? 

85.  Repeat  the  table  of  Circular  Measure.     Give  the  signs  which 
mark  its  denominations. 

86.  What  is  a  circle?  .  / 

87.  What  is  the  circumference  of  a  circle  ? 

88.  What  is  an  arc  of  a  circle  ?     A  quadrant  of  a   circle  ?     Its 
diameter  ?     Its  radius  ? 

89.  Can  you  repeat  a  Miscellaneous  .Table,    by  which  different 
sorts  of  merchandise  are  weighed  or  measured  ? 

90.  By  what  measure  is  land  measured  ? 

91.  By  what  measure  is  distance  reckoned  ? 

92.  By  what  measure  is  lumber  surveyed  ? 

93.  How  is  wood  measured  ?  V 

94.  How  is  coal  estimated  ? 

95.  How  is  railroad  freight  often  reckoned  ? 

96.  How  is  a  ship's  freight  reckoned  ? 


SECTioi^  vm. 

GENERAL  ARITHMETICAL  PRINCIPLES.  * 

1.  What  is  an  even  number  ? 

2.  What  is  an  odd  number  ? 

3.  What  is  a  prime  number  ? 

4.  Define  the  \fOvdi  prime. 

5.  What  is  a  composite  number  ? 

6.  What  is  the  only  even  prime  number  ? 

7.  When  are  numbers  said  to  be  mutually  prime? 

8.  Define  the  word  mutually, 

9.  What  is  a  Power? 

10.  What  is  a  Eoot  ? 

11.  How  is  the  power  of  a  number  expressed  ? 

12.  How  is  the  root  of  a  number  expressed  ? 

13.  A  number  is  what  power  of  itself? 

14.  A  number  is  what  root  of  itself? 


ARITHMETIC^  15 

15.   What  is  the  square  root  of  a  number?     The  cube  root? 
'  16.   Is  there  any  similarity  between  the  relations  of  a  composite 
number  and  its  factors,  and  a  number  and  its  rooty  or  a  number  and 
its  power?- 

17.  What  are  the  factors  of  a  number  ? 

18.  Is  a  number  a  factor  of  itself  ? 

19.  What  are  the  prime  factors  of  a  number  ? 

20.  What  is  meant  by  factoring  a  number  ? 

21.  What  numbers  are  divisible  by  2  ? 

22.  What  numbers  are  divisible  by  3  ?    4?     5?     6? 
23. '  What  numbers  are  divisible  by  9  ?     10  ?     11  ?     12  ? 

24.  What  general  principle  is  there  in  regard  to  the  divisibility 
of  numbers  ? 

25.  Define  the  word  problem, 

26.  Define  the  word  50^1^^1071. 

27.  What  is  the  meaning  of  each  of  these  words  in  Arithmetic? 

28.  What  is  meant  by  solving  a  problem  ? 

29.  What  are  ^n'me  factors  ? 

30.  How  may  a  number  be  factored,  so  that  all  its  factors  will 
be  prime  ? 

31.  What  are  composite  factors  ? 

32.  What  is  the  difference  between  the  above  kinds  of  factors  ? 

33.  How  are  the  composite  factors  obtained  from  the  prime? 

34.  What  is  a  common  divisor  ? 

35.  What  factors  must  a  common  divisoi*contain  ? 

36.  What  is  the  greatest  common  divisor? 

37.  What  factors  must  a  greatest  common  divisor  contain  ? 

38.  Are  there  other  names  for  the  greatest  common  divisor? 

39.  How  is  the  greatest  common  divisor  of  two  or  more  numbers 
found  ? 

40.  Give  an  example  illustrating  this. 

41.  When  the  numbers  can  not  readily  be  resolved  into  their 
prime  factors ,  how  may  the  greatest  common  divisor  be  found  ?  If 
there  are  more  than  two  numbers  ? 

42.  Explain  the  above  process,  and  give  an  example. 

43.  State  the  principles  upon  which  this  operation  is  based. 

44.  What  is  a  multiple  of  a  number  ? 

45.  What  factors  must  a  multiple  contain? 

46.  What  is  a  common  multiple  of  two  or  more  numbers  P^^ 

47.  What  factors  must  a  common  multiple  contain  ?      '^-^'^ 


16  ARITHMETIC. 

48.  What  IS  the  least  common  multiple  of  two  or  more  mimbers  ? 

49.  What  factors  must  the  least  common  multiple  contaia? 
^50.    How  is  the  least  common  multiple  found  ? 

51.  Upon  what  principle  does  this  process  depend? 

52.  Illustrate  this  process  by  an  example. 

53.  How  can  several  numbers  be  readily  resolved  into  their  prime 
factors  ? 

54.  How  in  the  last,  case  is  the  least  common  multiple  found  .^ 
Give  an  example. 

65.   When  any  of  the  given  numbers  are  measures  of  one  another, 
how  may  the  process 'be  shortened? 

56.  Name  other  methods  of  shortening  the  process,  if  any. 

57.  What  is  the  least  common   multiple  of  prime   or  mutually 
prime  numbers  ? 

58.  What  is  the  least  common  fliultiple  of  two  numbers  ? 


SECTION  IX. 

FRACTIONS. 

1.  What  is  a  Fraction  ?     Give  an  example. 

2.  What  is  a  Common  or  Vulgar  Fraction  ?     How  expressed  ? 

3.  What  does  the  number  below  the  line  indicate  ? 

4.  Define  the  word  denominator.  - 

5.  Why  is  the  number  below  the  line  so  called  ? 

6.  AVliat  is  the  number  above  the  line  called,  and  why  ? 

7.  What  are  the  numbers  above  and  below  the  line,  £aken  together, ' 
called  ? 

8.  How  does  the  expression  for  a  Common  Fraction  resemble  the 
expression  for  Simple  Division  ? 

9.  What  may  all. such  fractions  be  termed  m  respect  to  Division  ? 

10.  To  what  term  in  Division  does  the  value  of  a  Common  Frac- 
tion correspond  ? 

11.  Which  fundamental  operatipn,  then,  do  all  fractions  arise  from  ? 

12.  Give  the  fractional  expression  for  Division. 

13.  Which  number  is  the  divisor,  and  what  is  the  name  of  it  in  the 
fractional  expression  ? 

14.  Which  number  is  the  dividend,  and  what  is  the  name  of  it  in  the 
fractional  expression  ?  , 


ARITHMETIC.  17 

15.  Is,  then,  tlie  divisor  always  larger  than  the  dividend  ? 

16.  Name  the  general  principles  o?  Division  which  indicate  how 
the  multiplication  or  divisio-n  of  either  the  dividend  or  divisor,  or 
both,  affect  the  value  of  the  quotient? 

17.  How  does  multiplying  the  numerator  [dividend]  affect  the 
value  of  the  fraction  [quotient]  ? 

18.  How  does  dividing  the  numerator  [dividend]  affect  the  value 
of  the  fraction  [quotient]  ?  ' 

19.  How  does  multiplying  the  denominator  [divisor]  affect  the 
value  of  the  fraction  [quotient]  ? 

20.  How  does  dividing  the  denominator  [divisor]  affect  the  value 
of  the  fraction  [quotient]  ? 

21.  How  does  multiplying  or  dividing  both  numerator  [dividend] 
and  denominator  [divisor]  by  the  same  number,  affect  the  value  of 
the  fraction  [quotient]  ?    - 

22.  Always,  then,  an  increase  or  decrease  of  the  numerator  [divi- 
dend] how  in  either  case  affects  the  value  of  the  fraction  [quotient]  ? 

23.  Always,  then,  an  increase  or  decrease  of  the  denominator 
[divisor]  how  in  either  case  affects  the  value  of  the  fraction  [quo- 
tient] ? 

I     21.  What  is  a  Proper  Fraction  ?     Give  an  example. 
I     25.   What  is  an  Improper  Fraction  ?    Give  an  example. 

26.  What  is  a  Simple  Fraction?     Give  an  example. 

27.  What  is  a  Compound  Fraction  ?     Give  an  example. 

28.  What  is  a  Mixed  Number  ?     Give  an  example. 

29.  What  is  a  Complex  Fraction?     Give  an  example. 

30.  Define  the  word  complex. 

31.  What  is  the  reciprocal  of  a  number?         . 

32.  Define  the  word  reciprocal. 

33.  How  is  a  mixed  number  reduced  to  an  improper  fraction? 
Give  an  example. 

34.  How  is  an  improper  fraction  reduced  to  a  whole  or  mixed 
Dumber?    Give  an  example. 

35.'  Is  the  value  of  the  expression  changed  in  either  of  these 
reductions  ? 

36.  What  is  an  integer  ? 

37.  How  is  an  integer  reduced  to  the  form  of  a  fraction  ?  Is  the 
mlue  altered  ? 

•^)8.  How  is  a  fraction  reduced  to  its  lowest  terms  ? 
cJ9.  Meaning  of  lowest  terms  of  a  fraction  ? 


18  ARITHIMETIC. 

40.  Is  the  value  of  the  fraction  changed  by  this  reduction  ? 

41.  Upon  what  principle  does  this  reduction  to  lowest  terms 
depend  ? 

42.  By  what  two  methods  is  a  fraction  multiplied  by  a  whole 
number  ?     Give  examples. 

43.  What  general  principles  explain  the  reason  for  these  pro- 
cesses ? 

44.  What  determines  which  method  should  be  adopted  ? 

45.  What  is  the  product  of  a  fraction  multiplied  by  its  denomi- 
nator? 

46.  By  what  two  methods  is  a  mixed  number  multiplied  by  a  whole 
number?     Give  examples. 

47.  By  what  two  methods  may  a  fraction  be  divided  by  a  whole 
number?     Give  examples. 

48.  'What  determines  which  method  is  to  be  preferred  ? 

49.  What  general  principles  explain  this  process  ? 

60.  By  what  two  methods  may  a  mixed  number  be  divided  by  a 
whole  number?     Give  examples. . 

51.  How  is  one  fraction  multiplied  by  another  ?    Give  an  example. 

52.  How  many  times  smaller  is  a  fraction  than  the  integer  which^ 
the  numerator  represents  ? 

53.  Having,  then,  multiplied  any  fraction  by  the  numerator  of 
another  given  fraction,  is  the  product  too  large  or  too  small  for  the 
correct  result  of  multiplying  by  the  given  fraction  ? 

54.  If  too  large  or  too  small,  what  determines  how  many  times  so  ? 

55.  What,  then,  is  the  next  step  to  complete  the  multiplication  of  a 
fraction  by  a  fraction  ? 

bQ,   Give  an  example  and  illustrate  the  complete  process. 

57.  What  is  meant  by  cancelling^  in  Arithmetic  ? 

58.  Upon  what  general  principle,  already  treated  of,  does  cancel- 
ling depend  ? 

59.  AYhen  may  cancelling  be  used  to  advantage  ?  Give  an 
example. 

60.  In  cancelling,  when  should  the  quotient,  if  a  unit,  be  retained  ? 

61.  How  is  a  compound^ fraction  reduced  to  a  simple  one? 

62.  What  process  already  given  explains  this  operation  ? 

63.  How  is  a  whole  number  multiplied  by  a  fraction  ?  Give  an 
example. 

64.  How  is  a  mixed  number  multiplied  by  a  fraction  or  by  another 
mixed  number?    Give  an  example. 


ARITHMETIC.  19 

65.  How  IS  a  fraction  divided  by  a  fraction  ? 

66.  Having  divided,  a  fraction  by  the  numerator  of  any  given 
fraction,  is  the  quotient  thus  obtained  too  large  or  too  small  for 
a  correct   result  of  dividing   by  the   given  fraction?. 

67.  How  many  times  smaller  is  the  fraction  representing  the 
divisor  than  the  integer  repreapnted  by  its  numerator  ? 

68.  Having,  then,  divided  by  the  numerator  of  tiie  fraction,  what 
determines  how  many  times  too  small  or  too  large  the  quotient  ob- 
tained is  for  a  correct  result  of  dividing  by  the  given  fraction  ? 

69.  What,  then,  is  the  next  step  to  complete  the  division  of  one 
fraction  by  another  ? 

70.  Why  is  the  divisor  inverted  in  dividing  one  fraction  by 
another? 

71.  Upon  what  **  principles  of  division  "  are  the  processes  of  mul- 
tiplying and  dividing  fractions  by  one  another  explained  ? 

72.  What  is  the  reciprocal  of  a  fraction,  and  how  is  it  obtained? 

73.  How  Is  division  of  fractions  performed  when  the  denominators 
are  alike  ? 

74.  How  may  the  division  of  fractions  be  performed  when  the 
numerator  and  denominator  of  the  divisor  are  factors  of  the  corre- 
sponding terms  of  the  dividend? 

75;  How  is  a  whole  number  divided  by  a  fraction  ?  How  is  a 
inixed  number  divided  by  a  mixed  number?     Give  examples. 

76.  How  is  a  complex  fraction  reduced  to  a  simple  one  ?  Give  an 
example. 

77.  What  is  a  common  denominator? 

78.  How  is  a  common  denominator  for  several  fractions  obtained  ? 
Give  an  example. 

79.  Upon  what  **  principle  of  division  "  already  explained,  does  this 
process  depend  ? 

80.  AVhat  is  the  least  common  denominator? 

81.  How  is  a  least  common  denominator  for  several  fractions  ob- 
tained ?     Give  an  example. 

82.  What  principle  gives  the  reason  for  this  process  ? 

83.  Is  It  always  necessary  to  reduce  fractions  to  their  lowest  terms, 
before  proceeding  to  find  the  least  common  denominator? 

84   How  may  numerators  effractions  be  made  alike  ? 


20  ARITHMETIC. 

SECTIOI^  X. 

KEDUCTION,  ADDITION,  AND   SUBTRACTION   OF  FRACTIONS. 

1.  'How  is  a  fraction  of  a  higher  deijomlnation  reduced  to  a  frac- 
tion of  a  lower  denomination  ? 

2.  Give  an  example  of  the  above,  and  explain  the  process  of  mul- 
tiplying a  fraction  by  a  whole  number. 

3.  How  is  a  fraction  of  a  lower  denomination  reduced  to  a  fraction 
of  a  higher  denomination  ? 

4.  Give  an  example  of  the  above,  and  explain  the  process_of  divid- 
ing a  fraction  by  a  w^ole  number. 

5.  How  is  a  fraction  of  a  higher  denomination  reduced  to  whole 
number^  of  a  lower  denomination  ? 

6.  Give  an  example  of  the  above,  and  explain  the  process  of  re- 
ducing improper  fractions  to  whole  or  mixed  numbers; 

7.  When  one  quantity  is  a  part  of  another,  how  is  it  expressed 
fractionally?     Giv^  an  example. 

8.  Must  quantities  thus  compared  be  of  the  same  denomination  ? 
Why.P 

9.  How  are  whole  numbers  of  a  lower  denomination  reduced  to 
a  fraction  of  a  higher  denomination  ? 

10.  What  two  methods  of  this  reduction  ? 

11.  Explain  the  process  of  each. 

12.  In  what  cases  is  each  preferable? 

13.  Is  what  is  called  the  part  in  this  connection,  ever  equal  to  or 
greater  than  the  unit  with  which  it  is  compared  ? 

14.  What  kind  of  a  fraction  results  when  the  part  is  the  greater  ? 

15.  Can  numbers  of  different  denominations  be  added  directly? 

16.  When  numbers  are  added,  of  what  kind  or  denomination  is  the 
amount  ? 

17.  If  2  fifths  and  3  fifths  are  added,  what  is  the  result? 

18.  Can  2  sevenths  and  3  fifths  be  added  like  the  fifths  above  ? 

19.  When  different  fractions  are  to  be  adde4j  what  is  the  first  step 
to  be  taken  ?    Why  ? 

20.  What  is  the  next?    Explain  the  full  process. 

21.  How  can  the  process  of  adding  fractions  which  have  a  common 
numerator  be  performed  ? 

22.  To  what  is  this  last  process  equivalent? 


ARITHMETIC.  21 

23.  !Must  numbers  to  be  subtracted  from  one  anotlief,  be  of  tlie 
same  denomination  ? 

24:.  How  is  one  fraction  subtracted  from  another?  The  first  step? 
the  second  ?     Give  an  example. 

25.  How  can  the  process  of  subtracting  fractions  which  have  like 
numerators  be  performed  ? 

26.  To  what  is  this  contraction  equivalent  ? 

27.  How  is  a  whole  or  mixed  number- subtracted  from  a  whole  gr 
mixed  number  ? 

28.  Define  the  word  analysis. 

29.  How  is  an  example  in  Arithmetic  analyzed  ?  Give  an  exam- 
ple. 

80.   Can  all  examples  in  Arithmetic  be  performed  by  analysis  ? 
31.   Why  are  other  methods,  according  to  set  rules,  often  adopted  ? 


SECTION"  XI. 

DECIMAL  FRACTIONS. 

1.  What  is  a  Decimal  Fraction? 

2.  Define  the  word  dscimal,  axid  give  its  derivation. 

3.  When  we  speak  of  *'  Decimals ''  what  is  usually  meant  ? 

4.  What  may  be  the  denominator  of  a  Common  Fraction  ? 

5.  Of  what  class  of  numbers,  must  the  denominator  of  a  Decimal 
Fraction  always  be  ? 

6.  Are  all  the  principles  of  Common  Fractions  equally  applicable 
to  Decimals  ? 

7.  Is  the  denominator  of  a  decimal  fraction  expressed  ?    How  is 
it  known  ? 

8.  How  are  decimals  distinguished  from  whole  numbers  ? 

9.  What  is  the  decimal  j^oint  ? 

10.  What  is  the  first  place  at  the  right  of  the  decimal  point  named  ? 
Second?     Third? 

11.  Do  whole  numbers  and  decimals  increase  in  the  same  ratio 
towards  the  left  ? 

12..  Write  a  decimal  often  places,  and  numerate  it? 

13.  What  does  a  mixed  number  in  Decimal  Fractions  consist  of  ? 
Give  an  example. 

14.  In  which  direction  is  the  integral  part  numerated,  and  in  which 
direction  the  decimal  ? 


22  AEITIOIETIC. 

15.  "\Ybat  determines  the  name  and  value  of  a  figure  in  a  Decimal 
Fraction  ?     In  whole  numbers  ?  , 

16.  Moving  the  decimal  point,  in  a  number  towards  the  right,  has 
what  effect  ?     Towards  the  left,  what  effect  ? 

17.  In  what  two  ways  may  a  decimal  be  read  ?     Give  examples. 

18.  How  many  numerations  are  required  in  order  to  read  a  decimal  ? 
Explain. 

19.  What  is  the  effect  upon  the  value  of  a  (decimal  of  annexing 
one  or  more  ciphers  ? 

20.  What  principle  explains  the  above  result  ? 

21.N  How  is  the  value  of  a  decimal  affected  by  prefixing  one  or  more 
cii:>hers? 

22.  What  principle  gives  the  reason  for  the  above  result  ? 

23.  What  sort  of  a  fraction  results  from  placing  a  common  fraction 
at  the  right  of  a  decimal  ?     Give  an  example. 

24  How  are  decimal  fractious  written,  and  what  particular  care  is 
necessary?  ^ 

25.  How  can  any  ambiguity  of  meaning  be  avoided  in  reading  a 
whole  number  and  a  decimal,  as  200.003? 

26.  How  are  addition,  subtraction,  multiplication,  and  division  of 
decimals  performed  ? 

27.  How  are  decimals  wHtten  for  addition,  and  what  special  care  is 
necessary  ? 

28.  How  for  subtraction  ?     • 

29.  Where  does  the  decimal  point  fall  in  the  result,  in  e^ch  case  ? 

30.  If  there  are  more  decimal  places  in  the  subtrahend  than  in 
the  minuend,  what  must  be  done  ?  What  is  the  effect  upon  the  value 
of  the  decimal  ? 

31.  In  the  multiplication  of  decimals,  how  is  the  place  of  the  point 
in  the  product,  determined  ? 

32.  If  the  number  of  places  in  the,  product  is  less  than  the 
number  of  decimal  places  required,  what  is  to  be  done  ? 

33.  Illustrate  by  multiplying  by  decimals,  with  the  denominators 
written  out,  the  principle  on  which  the  pointing  off  in  decimal  multi- 
plication depends. 

34.  How  can  a  decimal  be  multiplied  by  10,  100,  1000,  &c.  ?  How 
multiplied  by  any  number  with  one  or  more  ciphers  at  the  right  ? 

35.  How  is  the  place  of  the  point  in  the  quotient  determined  in  the 
division  of  decimals? 


ARITHMETIC.  23 

36.  If  the  number  of  decimal  places  in  the  quotient  is  too  small  for 
the  requi^rem<int  of  the  decimal  point,  what  is  done  ? 

37.  Illustrate  by  dividing  by  decimals,  with  the  denominators  writ- 
ten out,  the  principle  on  which  the  pointing,  off  in  decimal  division 
depends. 

38.  Can  you  give  other  explanations  of  the  same? 

39.  If  there  are  more  decimal  places  in  the  divisor  than  in  the 
dividend,  what  must  be  done  ? 

40.  '\Yhen  there  is  a  remainder  in  dividing  decimals,,  how  may  the 
division  be  continued  ? 

41.  Can  decimal  division  always  be  thus  completed  ? 

42.  For  what  is  the  sign  +  used,  besides  to  express  addition  ? 

43.  How  can  a  decimal  be  divided  by  10,  100,  1000,  &c.  ?    How  . 
divided  by  any  number  with  one  or  more  ciphers  at  the  right? 

44.  Can  all  Decimal  Fractions  be  reduced  to  Common  Fractions, 
and  the  reverse  ? 

45.  How  is  a  Common  Fraction  reduced  to  a  decimal  ?  Give  an 
example. 

46.  What  principle  is  illustrated  in  this  process  ? 

47.  Is  every  decimal  virtually  a  Common  Fraction  ?     Illustrate. 

48.  How  may  the  above  process  of  reduction  be  proved  ? 

49.  How  are  whole  numbers  of  a  lower  denomination  reduced  to 
decimals  of  a  higher  denomination  ?     Give  an  example. 

50.  Compare  this  process,  by  giving  an  example  in  each,  with  re- 
ducing whole  numbers  of  a  lower  denomination  to  a  common  fraction 
of  a  higher  denomination. 

51.  Is  the  principle  in  each  case  the  same,  and  what  is  it? 

b2.  How  are  decimals  of  a  higher  denomination  reduced  to  whole 
numbers  of  a  lower  denomination  ? 

53.  Compare  this  process,  by  giving  an  example  in  each,  with  re- 
ducing common  fractions  of  a  higher  denomination  to  whole  numbers 
of  lower  denominations. 

54.  Is  the  principle  in  each  case  the  same,  and  what  is  it  ? 

bb.  What  is  the  cardinal  fact  to  be  always  borne  in  mind,  in  Ke- 
duction,  respecting  the  valur  of  the  quantity  reduced  and  the  result 
obtained  by  the  reduction  ? 


24  ARITHMETIC. 

SECTION  XII. 

FEDERAL  OR  U:XITED  STATES  BIONEY. 

1.  "VYhat  is  United  States  Money? 

2.  AViat  are  its  denominations  ? 

3.  Give  the  table. 

4.  In  what  denominations  is  it  usually  reckoned  ? 

5.  Upon  what  system  is  this  currency  based  ? 

6.  What  is  a  coin  ? 

7.  What  are  the  authorized  coins  of  United  States  money  .^ 

8.  Of  what  are  these  coins  severally  composed  ?         ^ 

9.  ^Yhsit  is  alloT/'?  *  , 

10.  For  what  is  alloy  used  ? 

11.  Explain  the  use  of  the  term  carat, 

12.  Pure  gold  is  how  many  carats  fine  ? 

13.  What  is  the  Standard  of  the  U.  S.  gold  and  silver  coin  ? 

14.  What  is  the  alloy  for  gold? 

15.  What  is  the  alloy  for  silver  ? 

16.  Of  what  are  the  smaller  coins  composed? 

17.  What  denomination  is  not  coined  ? 

18.  What  is  the  weight  of  the  eagle!? 

19.  What  is  the  weight  of  the  silver  dollar? 

20.  How  are  these  standards  regulated  ? 

21.  In  this  currency  what  denomination  is  regarded  as  the  unit  ? 

22.  How-  are  cents  and  mills  regarded  in  operations  in  Federal 
Money  ? 

23.  What  does  the  money  in  general  use  consist  of? 

24.  Have  bank  and  United  States  bills  any  actual  value,  or  do  they 
only  represent  value  ?  .     . 

25.  Why  are  they  ordinarily  used  in  preference  to  coin? 

26.  Why  are  bills  and  fractional  currency  now  used  in  the  United 
States? 

27.  Precisely  like  what  other  operations  are  the  processes  in  Fed- 
eral Money? 

28.  How  are  dollars  reduced  to  cents  ?  to  mills  ?    Give  examples. 

29.  How  are  cents  reduced  to  dollars?    Mills  to  dollars?     Give 
examples. 


J  ARITHMETIC.  *  25 

30.  What  should  be  particularly  observed  in  performing  opera- 
tions in  Decimal  Fractions  and  Federal  Money  ? 

31.  How  is  the  cost  of  a  number  of  things  found  from  the  price  of 
one  ? 

32.  How  is  the  price  of  one  found  from  the  cost  of  a  given  num- 
ber? 

33.  "VVbat  is  usually  understood  as  the  diiference  between  price 
and  co5^?* 

34.  How  is  the  number  of  things  found  when  the  price  and  cost 
are  known  ? 

35.  How ^s  the  cost  of  articles  sold  by  the  100  or  1000  found? 
Give  an  example. 

36.  What  Roman  numeral  indicates  hundreds  ?  what  thousands  ? 

37.  When  are  the  mills  taken  into  account  in  business  transactions  ? 

38.  Define  the  word  aliquot: 

39.  Give  a  table  of  aliquot  parts  of  a  dollar. 

40.  How  is  the  coSt  found  when  the  price  is  an  aliquot  part  of  a 
dollar  ?     Give  an  example,  and  analyze  it. 

41.  What  is  the  above  process  sometimes  called? 

42.  How  is  the  cost  found  when  the  number  of  articles  is  expressed 
by  a  compound  or  by  a  mixed  number  ?     Give  an  example. 

43.  What  is  Barter? 

44.  How  are  examples  in  Barter  usually  solved  ? 

45.  What  is  a  Bill  of  Goods  ?    Make  one  containing  ten  items. 

46.  What  is  meant  by  receipting  a  Bill? 


SECTION  xm. 


COMPOUND  NUMBERS. 


1.  Do  the  denominations  of  Compound  l!^umbers  increase  deci- 
mally? 

2:  Explain  how  the  denominations  of  Compound  Numbers 
increase,  by  an  example. 

3.  Do  the  single  numbers  expressing  the  denomination  increase 
decimally  ? 

'*  Price  refers  to  a  single  article,  cost  to  a  lot  of  articles. 
3 


26  ARITHMETIC. 

4.  Precisely  like  what  is  the  principle  of  adding,  subtracting, 
multiplying,  and  dividing  compound  numbers  ?  * 

5.  How  are  compound  numbers  written  for  addition? 

6.  With  which  denomination  do  we  commence  to  add  ? 

7.  Having  found  the  amount  of  the  first  denomination,  what  is 
the  next  step  ? 

8.  Give  an  example,  and  explain  the  complete  process. 

9.  What  is  the  proof  of  Compound  Addition  ? 

10.  What  care  must  be  observed  in  writing  the  numbers  of  a  single 
denomination  ?    In  adding  them  ? 

11.  How  is  any  fraction  in  the  amount  disposed  of? 

12.  May  the  amount  contain  a  denomination  higher  or  lower  than 
any  in  the  given  example  ? 

13.  How  are  compound  numbers  written  for  subtraction  ? 

14.  What  is  the  next  step  ? 

15.  If  the  number  of  any  denomination  in  the  minuend  is  smaller 
than  the  number  of  the  dent)mination  in  the  subtrahend,  what  is  to 
be  done  ? 

16.  Explain  the  full  process  of  Compound  Subtraction,  by  an 
example. 

17.  What  is  the  ^roo/ of  it? 

18.  In  cases  of  **  borrowing,"  in  Compound  Subtraction,  can  two 
of  the  next  higher  denomination  be  borrowed  as  well  as  one  ? 

19.  How  many  and  what  are  the  ^lethods  of  finding  the  difference 
between  two  dates  ?     Give  examples. 

20.  How  many  days  are  considered  a  month  ? 

21.  Can  compound  numbers  be  multiplied  by  compound  numbers  ? 

22.  What  kind  of  a  number  must  the  multiplier  always  be  both  in 
simple  and  in  compound  numbers  ? 

23.  Of  what  kind  is  the  product  always  in  Multiplication  ?    Why  ? 

24.  How  is  multiplication  of  compound  numbers  performed? 
Give  an  example. 

25.  How  is  this  multiplication  proved  ? 

26.  When  the  multiplier  is  a  composite  number  how  can  the 
work  be  performed  ? 

27.  When  the  multiplier  is  large  and  not  a  composite  number 
how  may  the  work  be  abbreviated  ?     Give  an  example. 

28.  Over  how  many  degrees  does  the  sun  appear  to  move  in 
twenty-four  hours?  Which  way?  How  many  degrees  then  in  one 
hour?  '  * 


ARITHMETIC.  27 

29.  Repeat  the  table  of  Longitude  and  Time. 

30.  How  is  the  difference  in  time  found  when  the  difference  in 
longitude  is  known  ?     Give  an  example. 

31.  If  your  watch  is  running  on  Boston  time,  and  you  should  visit 
-Chicago,  would  you  find  it  too  slow  or  too  fast  at  the  latter  place  ? 
How,  if  you  visited  London  ? 

32.  Which  has  the  hour  of  the  day  later  the  more  easterly  or  the 
more  westerly  place  ? 

33.  How  may  the  difference  in  longitude  be  found  when  the 
longitude  of  one  place  is  in  East,  and 'that  of  the  other  is  in  West 
longitude  ? 

34.  Can  one  compound  number  be  divided  by  another  ? 

-    35.   What  is  the  most  convenient  method  when  the  divisor  is  a 
compound  number  .^ 

36.  With  what  denomination  do  you  begin  in  dividing  a  compound 
number? 

37.  What  is  the  next  step  ? 

38.  Explain  the  complete  process  of  this  division  by  an  example  ? 

39.  When  the  divisor  is  a  composite  number,  how  can  the  division 
be  performed  ? 

40.  How  is  the  difference  in  the  longitude  of  two  places  found 
when  the  difference  in  time  is  known?     Give  an  example. 

41.  If  your  watch  is  running  on  true  time,  and  in  travelling  you 
find  it  indicates  too  slow  time,  do  you  infer  that  you  have  travelled 
East  or  AVest  ?    In  which  direction  if  you  find  your  watch  too  fast  ? 

42.  What  are  Duodecimals  ? 

43.  Define, the  word  duodecimal,  and  give  its  derivation. 

44.  Define  the  word  decimal,  and  give  its  derivation. 

45.  To  what  is  the  measure  of  Duodecimals  usually  applied  ? 

46.  How  are  its  denominations  indicated  ? 

47.  Which  denomination  is  regarded  as  the  unit  in  Dupdecimals  ? 

48.  How  are  addition  and  subtraction  of  Duodecimals  performed  ? 

49.  How  is  the  multiplication  of  Duodecimals  performed  ? 

50.  Can  one  duodecimal  be  multiplied  by  another? 

51.  Is  the  multiplier  in  this  case  actually  a  concrete  or  ab>5tract 
number  ?     Explain  the  process  by  an  example. 

52.  What  is  1  inch  in  Luiear  Measure?  In  Square  Measure? 
In  Cubic  Measure  ?    *  - 

53.  To  what  is  V  equivalent  ?  1"  ?  V"  ?  Explain  by  multiplying  1 
inch  jDy  1  inch  expressed  as  the  fraction  (y^^^)  of  a  foot. 


28  ARITHIVIETIC.  / 

54.  How  are  the  denominations  of  the  product  ascertained  In  the 
multiplication  of  Duodecimals  ?  Explain  find  Illustrate  by  an 
example,  in  which  the  inches  and  seconds  are  expressed  as  the 
fractions  of  a  foot. 

55.  Give  the  complete  process  of  multiplication  of  Duodecimals, 
illustrated  by  an  example. 

56.  How  is  the  division  of  Duodecimals  performed?  Give  an 
example. 

57.  How  is  the  division  performed  when  the  dividend  and  divisor 
are  both  compound  numbers  ?  . 

58.  Can  compound  numbers  be  reduced  to  whole  numbers  and 
decimals,  and  the  various  operations  of  adding,  subtracting,  mul- 
tiplying and  dividing,  performed,  and  then  the  product  reduced 
back  to  compound  numbers  ? 

59.  Is  the  latter  method  often  simpler  and  easier  ? 

,60.   In  what  country  are  all  the  Measures  on  the  scale  of  10  ? 

61.  Is  such  a  scale  an  advantage  ?     Why  ? 

62.  Could  ITumbers  themselves  be  founded  upon  a  scale  of  12, 
or  any  other  number  than  10  .^^  ' 


SECTION  XIY. 

PERCENTAGE  AND  INTEREST. 

1.  What  is  the  meaning  of    the   expression    per    cent. 7    Its 
derivation  ?  . 

2.  What  sign  is  used  to  indicate  it  ? 

3.  What  is  meant  by  rate  per  cent.  ?     Give  an  example.     What 
is  meant  by  per  annum  ? 

4.  What  is  meant  by  Percentage  ?     Give  an  example. 

5.  What  is  called  the  Base  of  Percentage?     Give^n  example. 

6.  In  what  ways  may  the  rate  per  cent,  be  expressed? 

7.  If  expressed  by  a  decimal  of  more  than  two  places,  how  must  the 
places  after  the  second  decimal  place  be  regarded  ? 

8.  How  is  the  Percentage  found  when  the  Base  and  Rate  are 
given  ?     Give  an  example. 

9.  How  is  the  rate  per  cent,  found  when  the  Base  and  Percentage 
are  given  ?     Give  an  example. 

10.  Do  the  last  two  operations  prove  each  other  ? 


.   ^  AEITHMETIC.  .  29 

11.  How  is  the  Base  found  when  the  Percentage  and  the  Rate  per 
cent,  are  given  ?     Give  an  example. 

12.  What  is  Interest  .5^ 

13.  ^Yhat  is  the  Principal  ? 

14.  What  is  the  Amount  ? 

15.  How  are  Percentage  and  Interest  related  .5* 

16.  In  Interest  what  corresponds  to  the  Base?  The  Percentage ? 
the  Rate  ? 

17.  How  is  the  rate  of  interest  usually  fixed  ? 

18.  What  is  a  higher  rate  than  legal  interest  called  ? 

19.  What  is  the  legal  rate  of  interest  in  New  England  and  in  many 
of  the  United  States  .^^  What  in  New  York?  What  in  various  other 
States  ? 

20.  ^Yhat  is  the  rate  of  the  United  States'  Courts  ?  Of  England  ? 
Of  France  ? 

21.  How  is  the  interest  of  $  1  at  6  per  cent,  found  for  any  given 
time? 

22.  How  is  the  interest  of  any  sum  at  6  per  cent,  found  for  any 
given  time  ?     Give  an  example,  and  analyze  the  process. 

23:  Do  you  know  or  can  you  reason  out  any  other  methods  of 
finding  the  interest  of  any  sum  at  6  per  cent,  for  any  given  time,  and 
what  are  they?    Illustrate  by  analyzing  examples. 

24.  How  is  interest  computed  on  pounds,  shillings,  and  pence  ? 

25.  How  can  interest  be  computed  on  any  foreign  money  ? 

26.  How  is  interest  found  at  any  other  rate  than  6  per  cent.,  first 
finding  it  at  6  per  cent.  ? 

27.  Can  you  reason  out  any  other  methods  of  finding  the  interest  of 
any  sum  for  any  given  tune  at  other  rates  than  6  per  cent.  ?  Illustrate 
by  an  example. 

28.  How  is  the  amount  of  any  sum  at  any  rate  for  any  time  ascer- 
tained? 

29.  What  is  a  Promissory  note  ? 

80.  What  is  the  person  who  signs  a  promissory  note  called  ? 

81.  What  is  the  person  to  whom,  or  to  whose  order,  it  is  to  be  paid 
called? 

82.  What  is  meant  by  indorsing  a  note  ?    Who  is  the  indorser  ? 

83.  What  is  meant  by  the  face  of  a  note  ? 

84.  Is  a  note  ever  signed  by  more  than  one  person  ?    What  is  such 
note  called  ? 

35.  What  is  a  negotiable  note? 
8* 


80  ARITHMETIC. 

36.  What  are  partial  payments  ? 

37.  How  is  the  interest  computed  on  promissory  notes,  when  no 
partial  payments  have  been  made  ? 

38.  How  is  such  interest  computed  when  partial  payments  have 
been  made  ? 

39.  What  is  to  be  dohe,  if  any  payment  does  not  equal  or  exceed 
the  interest  due  on  the  note  at  the  time  such  payment  is  made  ? 

40.  Why  is  this  provision  made  ? 

41.  Explain  the  full  process  of  computing  the  interest  on  notes,  on 
which  partial  payments  have  been  made? 

42.  When  a  note  is  settled  within  a  year  after  the  interest  com- 
menced, and  partial  payments  have  been  made,  how  is  it  customary  ta 
compute  the  interest  ? 

43.  What  other  method  of  computing  interest  on  promissory 
notes  ? 

44.  Considering  that  the  interest  of  $  1  at  6  per  cent,  is  1  mill  for 
6  days,  andl  cent  for  60  days,  how  may  the  interest  on  any  number 
of  dollars,  for  anytime,  be  computed  by  a  short  process? 

45.  What  is  the  most  simple  method  of  proof  in  interest? 

46.  In  reckoning  interest,  how  many  days  are  usually  considered  a 
month  ?     Is  this  exactly  correct  ? 

47.  In  what  cases  must  it  be  reckoned  on  the  exact  number  of 
days?  ' 

48.  If  the  interest  of  $1  is  5  mills  for  one  month,  what  is  the  inter- 
est for  one  fifth  of  a  month  or  6  days  ?  '  How  can  interest,  then,  easily 
be  reckoned  for  6  days  multiplied  by  10,100,  &c.  that  is,  for  2  months, 
20  months,  200  months  ? 

49.  What  is  Annual  Interest,  and  how  is  it  computed  ? 

50.  What  four  quantities,  or  elements,  do  we  deal  with  in  In- 
terest ? 

51.  How  many  are  given  and  how  many  required  in  each  problem  in 
Interest  ? 

52.  "When  the  Principal,  Interest,  and  Time  are  given,  how  is  the 
Rate  found  ?     Give  an  example,  and  analyze  it. 

53.  When  the  Principal,  Interest,  and  Rate  are  given,  how  is  the 
Time  found  ?     Give  an  example,  and  analyze  it. 

54.  How  is  the  Time  found  at  which  any  principal  will  double  itself 
at  any  rate  per  cent.  ? 

55.  When  the  Interest,  Time,  and  Rate  are  given,  how  is  the  Prin- 
cipal found?    Give  an  example,  and  analyze  it. 


I  ARITHMETIC.  31 

56.  When  the  Amount,  Rate,  and  Time  are  given,  how  istMiP¥nf=^ 
cipal  found  ?     Give  an  example,  and  analyze  it. 

57.  When  the   Principal,  Rate,  and, Time  are  given,  how  is  the 
Interest  found  ?  * 

58.  What  is  Compound  Interest? 

59.  How  often  may  the  interest  b.e  compounded? 

60.  Can  compound  interest  be  legally  collected  ?     Is  it  usury  ? 

61.  How  is  compound  interest  computed  ? 

62.  Is  compound  interest  at  3  per  cent /iaZ/*  as  much  as  at  6  per 
cent?    Why? 


SECTION  XV. 

DISCOUNT  AND  BANKING. 

1.  What  is  Discount? 

2.  What  is  meant  by  present  worth  ? 

3.  Is  it  correct  to  say  the  present  worth  of  a  sum  of  money  ?  Why  ? 

4.  What  is  2ideU'} 

5.  What  term  in  interest  does  the  debt  in  discount  correspond  to  ? 
What  the  present  worth  ?    What  the  discount  ? 

6.  How,  then,  is  the  present  worth  of  a  debt  found?     Give  an 
example. 

7.  How  is  discount  found  ?     Give  an  example. 

8.  To  what  is  the  interest  on  the  present  worth  equal  ? 

9.  What  is  a  Bank? 

10.  Meaning  of  the  word  incorporated^  ? 

11.  What  is  a  cAar^er  ? 

12.  What  is  the  Capital  Stock  of  a  Bank?    . 

13.  How  many  kinds  of  banks  are  there  ?     Describe  them. 

14.  What  threefold  purpose   do  banks  in  this  country    usually 
serve  ? 

15.  Who  control  the  affairs  of  a  bank? 

16.  What  are  the  officers  of  a  bank? 

17.  What  are  Bank  Bills  ? 

18.  What  is  a  Bank  Check? 

*Let  the  teacher  take  the  equation,I.=P.xE.XT.  and  illustrate  each  of  tho 
Problems  In  Interest. 


32  ARITHMETIC. 

19.  What  is  the  face  of  a  note  ?       '  ,     . 

20.  What  is  meant  by  the  maturity  of  a  note  ?  , 

21.  When  is  a  note  said  to  be  given  on  time  ? 

-    22.  When  is   such  a  note  legally   due,   that  is,   when  does   it 
mature? 

23.  What  are  days  of  grace  ?  ^ 

24.  When  a  note  becomes  due  on  Sunday,  or  any  legal  holiday, 
when  must  it  be  paid  ? 

25.  When  is  a  note  due  that  is  giveu  for  a  certain  number  of  days  ? 
When  given  for  a  certain  number  of  months,  when  due  ? 

26.  When  is  the  interest  paid  on  money  borrowed  of  a  bank? 
What  is  this  interest  called  ? 

27.  What  is  the  sum  of  money  received  by  the  borrower  from  the 
bank  called  ? 

28.  What  is  said  to  be  done  with  the  note  thus  used  at  a  bank? 

29.  How  is  bank  interest,  that  is,  bank  discount,  reckoned  ? 

30.  When  a  note  hearing  interest  is  discounted,  what  is  made  the 
base  for  discounting  ?    • 

31.  How  is  the  sum  found  for  which  a  note  must  be  written,  that 
the  proceeds  may  be  a  specified  sum?     Give  an  example. 

32.  The  difference  between  true  Discount  and  Bank  Disc(5unt  ? 
Illustrate. 


SECTION  XYI. 

INSURANCE,    STOCKS,    COMMISSION,  &C. 

1..  What  is  Insurance  ? 

2.  What  is  the  Premium,  and  how  is  it  computed? 

3.  Does  the  per  cent,  for  the  premium-vary  ? 

4.  The  more  hazardous  the  risk,  the  higher  or  lower  the'per  cent, 
premium? 

5.  Is  any  property  so  hazardous  that  it  is  not  easy  to  effect  an 
insurance? 

6.  By  whom  is  insuring  in  this  country  usually  carried  on  ?  Ever 
by  individuals  ? 

7.  What  is  meant  by  an  underwriter  ? 

8.  Can  you  name  any  of  the  benefits  of  insurance,  especially  to  a 
conmaercial  community?  ^  , 


ARITHMETIC.  83 

9.   AVliat  is  an  Insurance  Policy  ?     What  does  it  specify  ? 

10.  Is  property  usually  insured  for  its  full  value  ?     Why  ? 

11.  May  it  be  insured  at  more  than  one  office,  that  is,  by  more 
than  one  company  ?     When  ? 

12.  How  is  the  premium  computed  ? 

13.  Are  there  charges  for  insurance  besides  the  premium? 

14.  What  is  Capital  Stock  of  a?2?/ Company  ? 

15.  Name  some  of  the  different  kinds  of  incorporated  companies. 

16.  How  is  the  Capital  usually  divided  ?  .     ' 

17.  What  other  securities  are  called  stocks  ? 

18.  Meaning  of  securities  in  the  above  connection  ? 

19.  What  is  thenar  value  of  stock?        '  ' 

20.  What  is  the  market  value  of  stock,  and  is  it,  or  not,  usually  the 
same  as  the  par  value  ? 

21.  Give  an  example  of  stock  at  par.     Above  par.     Below  par. 

22.  What  are  dividends  ? 

23.  What  are  assessments  ? 

24.  When  the  dividends  are  large,  is  the  stock  likely  tb  be  above 
or  below  par  ?     Why  ? 

25.  When  the  assessments  are  large,  and  dividends  small,  is  the 
stock  likely  to  be  above  or  below  par  ? 

26.  How  are  dividends  and  assessments  computed  ? 

27.  How  is  the  market  value  of  stock  found,  when  the  par  value, 
and  premium  or  discount,  is  known  ? 

28.  How  is  the  number  of  shares  found  that  may  be  bought  for  a 
certain  sum,  when  the  discount  or  premium  is  known  ? 

29.  What  is  Commission  or  Brokerage  ? 

30.  What  is  an  Agent  ?  By  what  other  name  is  he  sometimes 
designated? 

31.  How  is  Commission  or  Brokerage  computed?  Give  an  ex- 
ample. 

32.  How  is  the  commission  or  brokerage  found,  the  rate  per  cent 
commission  being  given,  when  the  agent  is  to  take  his  pay  from  a 
certain  sum  and  invest  the  balance?    Give  an  example,  and  explain  it. 

33.  What  is  a  Tax? 

34.  By  whom  are  taxes  assessed  ?  ' 

35.  To  what  uses  are  taxes  applied? 

36.  How  is  a  tax  on  property  assessed? 

37.  How  is  a  tax  on  persons  assessed? 

38.  What  is  meant  by  a  poll  ? 


84  ARITHMETIC. 

39.  Wliat  are  the  two  cMef  divisions  of  property? 

40.  What  is  Real  Estate  ? 

41.  What  is  Personal  Property?: 

42.  AVhat  is  an  Inventory  P 

43.  Are  the  details  of  taxation  the  same  in  all  the  States?     What 
peculiarity  in  Vermont  ?     What  in  Connecticut  ? 

44.  What  is  the  Rule  for  assessing  taxes  in  Massachusetts  ? 

45.  What  is  the  rule  in  the  State  where  you  reside  ? 

46.  Are  there  other  taxes  besides  town  or  city,  county  and  state 
taxes  ? 

47.  What  is  an  Excise  Tax  ? 

48.  What  is  a  Stamp  Tax?        . 

49.  What  is  the  United  S"tates  Internal  Revenue  ? 

50.  What  **iabor-saving"  way  is  sometimes  employed  by  asses- 
sors in  computing  taxes  ? 

51.  In  calculating  a  tax-list,  what .  description  of  table  is  found 
very  useful  ? 

52.  What  kind  of  taxes  are  called  Customs  or  Duties  ? 

53.  For  what  are  these  taxes  laid  ? 

54.  What  are  the  only  ports  in  the  United  States  where  goods 
brought  from  foreign  countries  can  lawfully  be  landed,  called  ? 

55.  What  is  established  at  each  port  of  entry? 

56.  What  is  smuggling?     What  are  persons  engaged  in  it  sub- 
ject to? 

57.  What  is  Tonnage  ? 

58.  How  many  kindi  of  Duties  are  there,  and  what  are  they  called  ? 

59.  What  is  an  ad  valorem  duty  ? 

60.  What  is  a  specific  duty  ? 

61.  What  is  an  invoice  ?- 

^2.  How  are  ac?  ??aZorew  duties  computed  ?     Give  aft  example. 

63.  On  what  only  are  specific  duties  computed  ? 

64.  What  is  Leakage  ?    Breakage  ?     Draft  or  Tret  ? 

65.  What  is  Tare  ?     Gross  weight  ?     Ket  weight  ? 

6^.   How  are  specific  duties  computed  ?     Give  an  example. 


ARITHMETIC.    '  35 

SECTION  XYII. 

EXCHANGE  ANI>  EQUATION   OF  i»AYMENTS. 

1.  What  is  Exohange,  in  "commerce? 

2.  What  is  a  Draft  or  Bill  of  Exchange  ? 

3.  Who  is  the  Makier  or  Drawer  of  a  Bill  ? 

4.  Who  is  the  Drawee  of  a  Bill  ?    The  Payee  ? 

5.  Explain  the  operations  of  Exchange. 

6.  How  are  bills  made  payable  .P 

7.  When  is  a  bill  payable. aj5  siglii^    When  for  any  specified' 
time  ? 

8.  Define  the  word  negotiable. 

9.  What  is  a  negotiable  bill  or  note  ?. 

10.  When  a  person  sells  a  bill  of  exchange,  what  is  the  person 
who"  buys  it  called  ? 

11.  What  is  the  person  possessing  a  bill  at  any  time  called  ?    . 

12.  Who  is  the  indorser  of  a  bill  ? 

13.  In  what  manner  is  the  indorser  responsible  for  a  hWi? 

14.  What  is  it  to  accept  a  bill  ? 

15.  What  is  it  to.  protest  a  Bill  ? 

16.  What  officer  is  employed  to  protest  bills  ? 

17.  When  should  a  bill  regularly  be  presented  for  payment  ? 

18.  If  the  bill  is  not  paid  when  presented,  what  is  neeessary  to 
hold  indorsers  ? 

19.  What  are  Exports? 

20.  What  are  Imports  ? 

21.  When  is  the  balance  of  trade  in  our  favor?    When  against 
us  ? 

22.  By  what  must  the  deficiency  or  debt,  on  either  side,  be  made  up  ? 

23.  How  does  the  balance  of  trade  affect  Exchange  ? 

'24.  If  the  United  States  buy  more  of  England  than  they  sell  to 
England,  will  exchange  on  England  be  at  a  premium  or  discount  ? 
How,  if  they  sell  more  than  they  buy  ? 

25.  Will  the  variation  of  Exchange,  that  is,  the  premium  or  dis- 
count, ever  be  very  great  ?    Why  ? 

26.  Are  time  bills  of  exchange  subject  to  discount? 

27.  What  is  the  exchange  value,  in  the  United  States,  of  a  pound 
sterling  ?     The  intrinsic  or  commercial  value  of  a  pound  sterling  ? 

28.  Write  a  draft  or  bill  of  exchange  in  proper  form. 


36  ARITHMETIC. 

29.  An  order  or  bill  of  excliange  payable  In  a  country  where  It  is 
drawn,  is  called  what  ?     What  if  payable  in  a  foreign  coimtiy  ? 

30.  How  is  the  cost  of  a  bill  of  exchange  found  ?  Give  an  exam- 
ple. 

31.  How  is  the  face  of  a  bill  found  which  a  given  sjim  In  United 
States  Money  will  buy  ?  Give  an  example.  How,  when  United  States 
Currency  is  at  a  discount  ? 

32.  Define  the  word  equation, 

33.  What  is  Equation  of  Payments  ? 

34."  What  is  meant  by  the  equated  time  ? 

35.  What  is  meant  by  the  term  of  credit  ? 

36.  What  Is  meant  by  the  average  term  of  credit  ? 

37.  How  is  the  average  term  of  credit  for  several  bills  found  ? 
Give  an  example. 

38.  How,  from  the  average  term  of  credit,  is  the  equated  time 
found  ? 

39^  How  Is  the  time  most  conveniently  expressed  in  solving 
such  examples  ? 

40'.  Explain  a  method  of  equating  payments  by  means  of  reckoning 
the  interest.     The  reason  for  it. 

41.  What  Is  a  short  way  of  finding  the  interest  on  any  sum  for 
two  months  ?     How  then  for  one  month  ?  , 

42.  What  is  done  with  fractions  of  a  day  In  equation  of  payments  ? 

43.  Which  of  the  above  methods  of  equating  payments  is  prefer- 
able, and  why  ? 

44.  In  finding  the  average  date  of  debts  of  equal  terms  of  credit, 
from  what  date  may  the  interest  be  reckoned  ?  From  what  date  is  it 
most  conveniently  reckoned  ?    Why  ? 

45.  What  is  the  date  from  which  the  interest  on  several  bills  is 
reckoned  called  ? 

46.  How  is  the  equated  time  found  when  all  the  terms  of  credit  are 
equal,  but  begin  at  different  times  ?  Give  an  example,  and  analyze 
it. 

47.  What  is  the  maturity  of  a  bill  or  note  ? 

48.  When  the  terms  of  credit  are  unequal,  and  begin  at  different 
times,  bow  is  the  equated  time  found  ?  Give  an  example,  and  analyze 
it. 

49.  How  does  the  last  case  differ  from  the  one  which  precedes 
it?    • 

60.   How  is  the   equated  time  found  for  paying  the  balance  of  an 


ARITHMETIC.  37 

account  which  has  both  debit  and  credit  entries  ?'    Give  an  example, 
and  analyze  it. 

51.  Where  the  larger  interest  arises  on  the  smaller  side  of  the  ac- 
count, what  is  the  result  ?     What  must  then  be  done  ? 

52.  What  determines  the  choice  of  focal  date  ? 

53.  What  is  the  principle  upon  which  Equation  of  Payments  is 
based? 


SECTION"  XVIII. 

PROFIT  AND  LOSS,   AND  PARTNERSniP. 

1.  Explain  the  term  **  Profit  and  Loss." 

2.  What  is  absolute  gain  or  loss  ? 

3.  What  is  percentage  of  gain  or  loss,  and  on  what  is  it  comput- 
ed? 

4.  How  is  absolute  gain  or  loss  found  ?  Give  an  example,  and 
analyze  it. 

5.  How  is  the  per  cent,  of  gain  or  loss  found  when  the  cost  and 
selling  price  are  given  ?^    Give  an  example,  and  analyze  it. 

6.  How  can  the  selling  price  be  found  when  the  cost  and  gain  or 
loss  are  known  ?     Give  an  example,  and  analyze  it. 

7.  How  can  the  cost  be  found  when  the  selling  price  and  per  cent, 
of  gain  or  loss  are  given  ?     Give  an  example,  and  analyze  it. 

8.  How  is  the  per  cent,  of  gain  or  loss  found,  when  it  is  proposed 
to  sell  goods  at  any  given  price  ?  How  when  only  the  per  cent,  gained 
or  lost,  if  sold  at  a  given  price,  are  known?  Give  an  example,  and 
analyze  it. 

9.  How  may  the  marking  price  of  goods  be  found,  so  that  the 
merchant  may  fall  a  certain  per  cent.,  and  yet,  sell  the  goods  at  cost, 
or  at  a  certain  per  cent,  gain  or  loss  on  the  cost  price  ?  Give  an  ex- 
ample, and  analyze  it. 

10.  What  is  Partnership  ? 

1 1.  AVhat  is  a  Partnership  Company  often  called  ? 

12.  What  is  meant  by  an  active  and  what  by  a  silent  partner? 

13.  What  is  the  Capital  or  Stock  ? 

14.  How  are  the  profits  and  losses  divided? 

15.  What  is  the  difference  between  a  Partnership  Company  and  an 
Insurance,  Manufacturing,  or  Railroad  Company  ? 

4 


38  ARITHMETIC. 

16.  By  what  authority  is  this  difference  established,  and  what  au- 
thority defines  the  privileges  and  responsibilities  of  each  ?      «. 

17.  What  is  a  c/iar^er  ? 

18.  Can  you  tell  the  object  of  compelling  certain  companies  to  ob- 
tain charters  of  incorporation  ? 

19.  How  is  each  partner's  share  of  the  gain  or  loss  found?  What 
other  method  ?     Give  an  example. 

20.  What  is  the  proof  of  the  above  process  ? 

21.  How  is  each  partner's  share  of  gain  or  loss  ascertained,  when 
, their  capital  is  employed  unequal  times?     Give  an  example. 

22.  Upon  what  principle  is  the  above  computation  based  ? 

23.  Can  all  problems  which  come  under  the  head  of  Percentage  be 
performed  by  analysis  ?  What  is  the  advantage  of  special  rules  in  the 
various  cases  ? 


SECTION  XIX. 

RATIO  AND  PROPORTION. 

/       1.  AVhat  is  Ratio  ? 

2.  How  is  the  ratio  between  two  numbers  usually  expressed? 

3.  What  are  the  quantities  compared  called  ? 

4.  What  is  the  first  term  called?  The  second?  The  two  to- 
gether ? 

5.  Which  term  is  considered  the  divisor  ? 

6.  Is  the  other  term  ever  considered  the  divisor? 

7.  Which  is  sometimes  called  the  English  and  which  the  French 
method  ? 

8.  Which  is  called  a  direct  and  which  a  reciprocal  ratio  ? 

9.  Considering  one  term  of  the  ratio  as  the  divisor  and  the  other  as 
the  dividend,  what  general  principles  apply  to  them  as  affecting  the 
quotient? 

10.  How,  then,  is  the  ratio  affected  by  multiplying  the  antecedent 
[dividend]? 

11.  How  is  the  ratio  affected  by  dividing  the  antecedent  [dividend]  ? 

12.  How  is   the   ratio   affected   by  multiplying    the    consequent 
[divisor]  ? 

13.  How    is    the    ratio    affected    by    dividing    the    consequent 
[divisor]  ? 


ARITHMETIC.  99 

14.  ITow  is  tlie  ratio  affected  by  multiplying  or  dividing  botli 
antecedent  [dividend]  and  consequent  [divisor]  by  the  same  nmnber  ? 

15.  How  are  the  antecedent,  consequent,  and  ratio  related  to  each 
other  .^ 

16.  What  is  a  simple  ratio  ? 

171   What  is  a  compound  ratio  ? 
ISy^Vhat  is  the  value  of  a  compound  ratio  ? 
~^  ly.   What  is  Proportion  ? 

20.  IIow  is  a  proportion  indicated  ?     How  read  ? 

21.  When  are  four  numbers  in  proportion  ? 

22.  What  are  the  first  and  last  terms  called? 

23.  What  are  the  two  middle  terms  called? 

24.  A¥hat  are  the  first  and  third  terms  called  ? 

25.  What  are  the  second  and  fourth  terms  called? 

26.  To  what  is  the  product  of  the  extremes  equal? 

27.  What  is  meant  by  the  "  Rule  of  Three"  ? 

28.  How  many  terms  of  a  proportion  must  be  given  ? 

29.  From  the  given  terms,  how  can  the  remainder  of  the  proportion 
be  found  ? 

80.  The  product  of  the  extremes  divided  by  either  mean,  will 
give  what  ? 

31.  The  product  of  the  means,  divided  by  either  extreme,  will  give 
what  ? 

32.  Give  ai|  example,  and  illustrate  the  above. 

33.  Which  ^air5  of  terms  of  a  proportion  may  be  multiplied  or 
divided  by  the  same  number  without  destroying  the  proportion  ? 

34.  What  is  the  effect  of  multiplying  or  dividing  the  four  terms  of 
a  proportion  by  the  same  number  ? 

85.  In  how  many  orders  may  the  four  terms  of*  a  proportion  be 
written,  and  the  numbers  still  be  in  proportion  ?  Give  |n  example  il- 
lustrating this. 

36.  What  is  meant  by  a  mean  proportional  ? 

37.  How  is  a  mean  proportional  foun^l?     Give  an  example. 

38.  What  is  a  third  proportional  ?     How  found  ? 

30.  Of  what  kind  must  two  of  the  three  numbers  given  in  a  simple 
proportion  be  ?     Of  what  kind  the  other  ? 

40.  How  is  an  example  in  simple  proportion  performed  ?  Explain 
by  giving  an  example. 

41.  By  what  other  method  can  every  example  in  proportion  be 
solved  ?    Solve  an  example  by  proportion  and  then  by  analysis. 


40  ARiTrorETic. 

42.  AVhat  IS  Compound  Proportion  ? 

43.  How  is  an  example  in  compound  proportion  p'^fformcd? 
Explain,  by  giving  an  example. 

44.  Can  every  example  in  compound  proportion  be  solved  by 
simple  proportion  ?     Give  an  example. 

45.  By  what  other  method  than  those  mentioned  can  all  examples 
in  compound  proportion  be  solved  ?  Give  an  example,  and  solve  it 
by  compound  proportion,  by  simple  proportion,  and  then  by  analysis. 

46.  Why  are  the  methods  given  in  proportion  used  in  preference 
to  analysis  .»* 


SECTION  XX. 

ALLIGATION. 

1.  What  is  Alligation?     Of  what  two  kinds? 

2.  What  is  Alligation  Medial  ?     Give  an  example. 

3.  How  is  the  price  of  a  mixture  found,  when  the  quantities 
and  prices  of  the  articles  are  given  ?     Give  an  example. 

4.  What  is  Alligation  Alternate  ? 

5.  When  the  prices  of  several  kinds  are  given,  how  is  it  ascer- 
tained how  much  of  each  kind  may  be  taken  to  form  a  compound  of  a 
proposed  medium  price  ?  Give  an  example,  and  explain  the  process 
by  analysis. 

6.  May  different  answers  be  obtained  and  yet  all  be  correct? 

7.  Explain  a  method  where  the  quantities  are  at  first  assumed, 

8.  When  the  price  of  each  of  the  simples,  the  price  of  the  coin- 
poimd,  and  the  quantity  of  one  kind  are  given,  how  is  the  quantity 
of  the  other  simples,  which  may  be  taken,  found?     Give  an  example. 

9.  When  4he  prices  of  the  several  simples,  the  price  of  the  com- 
pound, and  the  entire  qua,ntity  in  the  compound  are  given,  how  is  it 
ascertained  how  much  of  each  simple  may  be  taken?  Give  an 
example. 


SECTio:Nr  XXL 

INVOLUTION  AND  EVOLUTION. 

1.  What' is  a  Power  of  a  Number? 

2.  What  is  Involution  ?     Give  the  derivation  of  the  word. 


ARITHMETIC.  41 

3.  The  number  to  be  involved  is  what  power  of  itself?    What  is 
it  in  respect  to  other  powers  of  itself? 

4.  What  is  the  index  or  exponent  of  a  power  ? 

5.  How  is  a  number  involved  to  any  required  power  ?     Give  an 
example. 

6.  Plow  is  the  involution  expressed?     Give  an  example. 

7.  How  is  a  common  fraction  involved?     A  mixed  number? 

8.  How  is   a  decimal  fraction  involved?     How  many   deeimal 
places  in  any  required  power  of  a  given  decimal  ? 

9.  What  is  the  sign  of  inequality,  and  its  use  ? 

10.  What  are  the  powers  of  1  ?  In  respect  to  size,  what  relation 
do  powers  of  numbers  greater  than  unity  bear  to  the  numbers  ?  The 
power  of  numbers  less  than  unity  ? 

11.  In  involving  a  number  what  determines  how  many  multiplica- 
tions are  required  ? 

12.  To  what  power  is  the  product  of  two  or  more  given  powers 
equal  ? 

13.  How  is  a  quantity  involved  that  is  already  a  power?  Give  an 
example. 

14.  How  is  the  power  of  any  number  divided  by  any  other  power 
of  the  same  number  ? 

15.  The  product  of  two  given  numbers  consists  of  how  many 
figures  ?     How  is  this  shown  ? 

16.  The  square  of  a  number  consists  of  how  many  figureiS  ?  The 
cube  or  third  power,  of  how  many  ?     The  fourth  power  ?  ' 

17.  The  square  of  units  may  consist  of  how  high  an  order  of 
figures  ?     The  square  of  tens,  of  how  high  an  order?     How  low? 

18.  What  is  Evolution?    Define  the  word,  and  give  its  derivation. 

19.  What  is  given  and  what  required  in  Involution  ? 

20.  What  is  given  and  what  required  in  Evolution  ? 

2 1 .  What  is  the  Root  of  a  number  ? 

22.  What  is  the  relation  of  powers  and  roots  to  each  other? 

23.  How  many  methods  are  there  of  indicating  a  root,  and  what 
are  they  ? 

24.  In  what  consists  the  process  of  Evolution  or  extracting  roots  ? 

25.  How  is  a  number  involved,  or  raised  to  a  certain  power  ?  How 
evolved,  or  the  root  extracted  ? 

26.  How  are  fractional  indices  found  ? 

27.  How  is  the  power  and  root  of  a  number  indicated  at  the 
same  time  ?    Any  other  way  ?     Give  examples. 

4* 


43  AEITHMETIC. 

28.  Can  all  num'bers  be  involved  to  any  required  power?  Can  all 
numbers  be  evolved'?  * 

29.  What  is  a  perfect  power  ?     An  imperfect  power  ? 

30.  What  is  a  rational  number  ?  An  irrational,  radical,  and  surd 
number  ? 

31 .  What  is  every  root  of  1  ?  In  respect  to  size,  what  relation  do  the 
roots^  of  numbers  greater  than  unity  bear  to  their  powers  ?  The  roots 
of  numbers  less  than  unity  ? 


SECTioN^  xxn. 

SQUARE    AKD   CUBE  ROOT. 

1.  What  is  it  to  extract  the  Square  Root  of  a  number? 

2.  Of  how  many  figures  does  the  square  or  second  power  of  a 
number  consist?  Of  how  many  the  square  root  of  a  number? 
Explain. 

3.  Of  what  order  of  figures  must  the  square  of  units  be?  Of 
tens  ?     Give  an  example. 

4.  In  a  number  of  three  figures,  of  how  many  figures  will  the 
root  consist  ? 

5.  How  is  the  number  of  figures  of  which  the  root  will  consist 
ascertained  and  indicated  ? 

6.  In  what  order  of  figures  must  the  square  of  tens  be  found  ? 

7.  What,  then,  is  the  first  step  in  extracting  the  root  of  a  number 
of  three  figures  ? 

8.  Having  found  the  tens  figure  of  the  root,  how  is  the  number, 
from  which  the  unit  figure  is  obtained,  found  ?  How  is  the  unit  fi<mre 
itself  of  the  root  then  found?  What  is  then  done  to  complete  the 
process  ? 

9.  Perform  an  example,  and  illustrate  the  process  by  a  diagram. 

10.  Why  is  the  root  already  found  doubled  for  a  trial  divisor  ? 

11.  Why,  in  completing  the  square,  do  we  suppose  additions  to  be 
made  to  two  sides  only  ? 

12.  What  does  the  last  root  figure  found  show  in  respect  to  the 
additions  to  be  made  to  the  square? 

13.  AVhy  is  each  root  figure  placed  at  the  right  of  the  trial  divisor? 

14.  Why  is  the  figure,  thus  obtained  f9r  the  root  sometimes  too 
great? 


ARITHMETIC.  43 

15.  What  does  the  trial  divisor  show  in  respect  to  the  square? 
The  true  divisor  show  ? 

16.  The  same  reasoning  applying  to  all  cases  of  extracting  the 
square  root,  what  general  direction  for  this  process  may  be  given  ? 

17.  Does  the  left-hand  period  always  consist  of  two  figures  ? 

18.  When  the  trial  divisor  produces  a  quotient  too  large,  what  is 
to  be  done  ? 

19.  What  method  of  proof  is  there  of  this  process  ? 

20.  How  is  the  square  root  of  a  mixed  decimal  number,  or  of  a 
decimal,  extracted  ? 

21.  Where,  in  the  last  instance,  must  the  first  point  be  placed  in 
indicating  the  periods  ?  If  the  right-hand  period  is  incomplete,  what 
is  done  ? 

22.  In  pointing  ofF,  to  ascertain  the  number  of  figures  in  the  root, 
where  should  the  first  point  always  be  placed,  in  whole  numbers,  and 
in  decimal's  ? 

23.  What  is  an  approximate  root,  and  how  is  it  extracted  ? 

24.  How  is  the  square  root  of  a  common  fraction  or  mixed  number 
extracted  ?     If  either  term  is  an  imperfect  square,  what  may  be  done  ? 

25.  What  is  an  angle  ?     What  the  vertex  of  an  angle  ? 

26.  How  is  an  angle  designated  ?  If  by  three  letters,  which  stands 
at  the  vertex  ? 

27.  When  is  it  necessary  to  use  three  letters  in  designating  an 
angle  ? 

28.  What  is  the  measure  of  an  angle  .^ 

29.  What  is  a  right  angle? 

30.  When  are  lines  perpendicular  to  each  other? 

31.  AVhat  is  an  acute  angle? 

32.  What  is  an  obtuse  angle? 

33.  What  is  an  oblique  line  ? 

34.  What  is  a  chord  ?    A  tangent  ? 

35.  What  is  a  triangle?     Derivation  of  the  word. 

36.  What  is  a  right-angled  triangle  ? 

37.  What  is  the  hypothenuse  of  a  right-angled  triangle?  The 
base  ?     The  perpendicular  ? 

38.  What  is  an  equilateral  triangle  ? 

39.  What  is  a  vertical  angle  ? 

40.  What  is  an  isosceles  triangle  ? 

41.  What  is  a  scalene  triangle? 

42.  What  is  a  reotangle  ? 


4:4  ARITHMETIC. 

43.  What  is  a  square?    A  diagonal? 

44.  What- is  an  inscribed  figure?  , 

45.  What  is  a  circumscribed  figure  ? 

46.  What  is  a  circumscribed  circle? 

47.  What  is  an  inscribed  circle  ? 

48.  To  what  is  the  sum  of  the  three  angles  of  a  plane  triangle 
equal  ? 

49.  To  what  is  the  sum  of  the  acute  angles  in  a  right-angled  tri- 
angle equal  ? 

-50.   Which  angles  of  an  isosceles  triangle  are  equal  each  to  the 
other  ? 

51.  How  are  the  angles  of  an  equilateral  triangle  related  to  each 
other  ? 

52.  If  two  angles  of  a  triangle  are  equal,  how  do  the  sides  opposite 
to  the  angles  compare  ? 

•    53,  If  a  perpendicular  is  drawn  from  the  vertex  of  an  equilateral 
or  isosceles  triangle  to  the  base,  how  does  it  divide  the  base  ? 

54.  IIow  does  a  diagonal  divide  a  square  ? 

55.  When  are  two  triangles  similar  ?     How  are  their  sides  related  ? 

56.  If  one  angle  of  a  right-angled  triangle  is  30°,  the  side  opposite 
that  angle  will  bear  what  relation  to  the  hypothenuse  ? 

57.  What  is  the  ratio  of  the  diameter  of  a  circle  to  its  circumfer- 
ence ?  What  follows  from  this  when  either  the  diameter  or  circum- 
ference of  a  circle  is  known? 

58.  How  is  the  area  of  a  circle  found  when  the  diameter  is  known  ? 
The  diameter  found  when  the  area  is  known  ? 

59..  What  is  the  ratio  of  the  areas  of  two  circles  to  each  other? 

60.  In  a  right-angled  triangle,  to  what  is  the  square  of  the 
hypothenuse  equal  ?  To  what  is  the  square  of  either  side  about  the 
right  angle  equal  ? 

61.  What  is  the  relation  of  the  squares  of  lines  of  different  lengths  ? 

62.  AVhat  is  a  cube  ? 

63.  What  is  it  to  extract  the  Cube  Root  of  a  number? 

64.  Of  how  many  figures  does  the  cube  of  a  number  consist  ?  how 
many,  then,  the  cube  root  of  the  number  ? 

Go,  In  a  number  of  five  figures,  of  how  many  will  the  cube  root 
consist  ?    How  is  this  ascertained  and  indicated  ? 

6Q.   In  what  order  of  figures  must  the  cube  of  tens  be  sought? 

67.  What  is  the  first  step  in  extracting  the  cube  root  of  a  number 
consistmg  of  five  figures  ? 


ARmOIETIC.  45 

68.  Having  found  the  tens  figure  of  the  root.  Low  is  tLe  number 
from  which  the  unit  figure  must  be  obtained,  found?  IIow  is  the 
unit  figure  itself  of  the  root  then  found?  What  is  then  done  to  com- 
plete the  process  ?  Extract  the  cube  root  of  a  number  of  five  figures, 
and  illustrate  the  process  by  a  diagram. 

G9.  In  getting  the  trial  divisor,  why  do  you  square  the  root  figure 
already  found?  Why  do  you  annex -two  ciphers  to  this  square  and 
then  multiply  the  result  by  three  ? 

70.  What  does  the  last  root  figure  found  show  in  respect  to  the 
additions  to  be  made  to  the  cube  ? 

71.  Why  do  you  annex  one  cipher  to  each  root  figure  multiplied 
by  the  root  previously  found,  and  then  multiply  this  result  by  three  ? 

72.  What  does  the  trial  divisor  show  in  respect  to  the  cube  ?  The 
true  divisor  sh«'>w? 

73.  The  same  reasoning  applying  to  all  cases  of  extracting  the  cube 
root,  give  the  full  directions  for  extracting  the  cube  root  of  any 
number. 

74.  What  IS  the  proof  of  this  process  ? 

75.  Can  the  cube  root  of  decimals,  mixed  decimal  numbers,  com- 
mon fractions,  and  mixed  numbers,  be  extracted?     How? 

76 .  When  are  rectangular  bodies  similar  ?  What  is  the  ratio  between 
similar  bodies  ? 

77.  Give  the  full  directions  for  extracting  a  root  of  any  degree. 


SECTION  xxm. 

PROGRESSIONS,   ANNUITIES,    AND   PERMUTATIONS. 

1.  When  is  a  series  of  numbers  said  to  be  in  Arithmetical  Pro- 
gression ?     Give  an  example. 

2.  How  many  kinds  of  series  are  there  in  Arithmetical  Progres- 
sion, and  what  are  they  ? 

3.  What  are  the  terms  of  a  series  ? 

4.  What  are  the  first  and  last  terms  of  a  series  called?    What 
the  other  terms  ? 

5.  AVhat  is  meant  by  the  common  difference  ? 

6.  How  many  particulars  are  considered  in  examples  in  Arith- 
metical Progression  ?    What  are  they  ? 


46  ARITHMETIC. 

7.  How  many  of  tlie  terms  must  be  given  in  order  to  find  tlie 
others  ?  ' 

8.  How  is  an  ascending  series  formed  ?     A  descending  series  ? 

9.  How  is  the  last  term  found,  when  the  first  term,  common  differ- 
ence, and  number  of  terms  are  given?     Give  an  example. 

10.  How  is  the  common  difference  found,  when  the  extremes  and 
number  of  terms  are  given  ?     Give  an  example. 

11.  How  is  the  number  of  terms  found,  when  the  extremes  and 
common  difference  are  given?     Give  an  example. 

12.  How  is  the  sum  of  the  series  found,  when  the  extremes  and 
number  of  terms  are  given  ?     Give  an  example. 

13.  What  constitutes  a  series  in  Geometrical  Progression  ? 

14.  How  many  kinds  of  series  in  a  Geometrical  Progression,  and 
what  are  they  ? 

15.  How  many  particulars  are  considered  in  problems  in  Geomet- 
rical Progression  ?  What  are  they  ?  How  many  of  them  must  be 
given  to  find  the  others  ? 

16.  How  is  an  ascending  series  formed  ?     A  descending  series  ? 

17.  How  is  the  last  term  found  when  the  first  term,  ratio,  and 
number  of  terms  are  given?     Give  an  example. 

18.  How  is  the  ratio  found  when  the  extremes  and  number  of  terms 
are  given?     Give  an  example. 

19.  How  is  the  sum  of  the  series  found  when  the  extremes  and 
ratio  are  given  ?     Give  an  example.      ^^ 

20.  What  is  an  Annuity  ?  A  certain  Annuity  ?  A  perpetual  Annuity  ? 

21.  What  is  an  annuity  in  arrears  ?  '  ij 

22.  What  is  meant  by  the  amount  of  an  annuity  ?  '^ 

23.  How  is  the  amount  of  an  annuity  in  arrears,  at  simple  interest, 
found  ?     Give  an  example. 

24.  How  is  the  amount  of  an  annuity  in  arrears,  at  compound  in- 
terest, found  ?     Give  an  example. 

25.  How  is  the  present  worth  of  an  annuity  certain^  at  compound 
interest,  found?     Give  an  example. 

26.  How  is  the  present  worth  of  an  annuity  perpetual  found? 

27.  What  is  Permutation?  How  is  the  number  of  permutations 
of  a  certain  number  of  things  found  ?     Give  an  example. 

28.  How  is  the  number  of  arrangements  that  can  be  made  of  a 
certain  number  of  things  taken  in  sets,  as  2  and  2,  3  and  3,  &c., 
found? 

29.  How  is  the  number  of  comhiyiations  of  any  number  of  things, 
in  sets  of  2  and  2,  3  and  3,  &c.,  found? 


AKITHMETIC.  4:7 


SECTIOI^J"  XXIV. 


MENSUIIATION. 


1 .  What  is  Mensuration  ? 

2.  What  are  parallel  lines  ? 

3.  What  is  an  oblique  angle  ? 

4.  AVhat  is  a  triangle  ?    Its  ha^e  ?    Its  altitude  ? 

5.  How  is  the  area  of  a  triangle  found? 

6.  What  is  a  quadrilateral  or  quadrangle?      Of  how  many  kinds? 

7.  What  is  a  trapezium ?     Trapezoid?     Parallelogram? 

8.  What  is  the  diagonal  of  a  figure  ? 

9.  What  the  altitude  of  a  trapezoid?    Of  a  parallelogram? 

10.  How  is  the  area  of  a  trapezium  found  ? 

1 1 .  How  is  the  area  of  a  trapezoid  found  ? 

12.  How  is  the  area  of  a  parallelogram  found  ? 

13.  AVhat  is  a  polygon  ?     The  perimeter  of  a  polygon  ? 

14.  What  are  the  names  of  some  of  the  different  polygons? 

15.  How  is  the  area  of  a  circle  found?    ^ 

16.  What  is  a  prism  ? 

17.  What  is  a  cylinder  ? 

18.  How  is  the  surface  of  a  prism  or  cylinder  found  ? 

19.  How  are  the  solid  contents  of  a  prism  or  cylinder  found  ? 

20.  AVhat  is  a  pyramid  ?     Its  vertex  ?     Slant  height  ? 

21.  What  is  a  cone  ?     The  altitude  of  a  pyramid  or  cone  ? 

22.  How  are  the  solid  contents  of  a  pyramid  or  cone  found  ? 

23.  What  is  the  frustum  of  a  pyramid  or  cone?     How  are  its  solid 
contents  found  ? 

24.  What  is  a  sphere  or  globe  ?     Its  diameter? 

25.  How  is  the  area  of  the  surface  of  a  sphere  found  when  the  cir- 
cumference and  diameter  are  known  ? 

26.  How  is  the  volume  or  solid  contents  of  a  sphere  or  globe 
found  when  the  surface  and  diameter  are  known  ? 


THE  END. 


EATON'S  SERIES  "of  ARITHMETICS. 


"The  High-School  Arithmetic  Is  all  that  could  reasonably  be  de- 
sired."—  J.  D.  PiiiLBRiCK,  Superintendent  of  Public  Schools,  Boston, 

**The  Common-School  Arithmetic  seems  to  combine  all  the  essen- 
tial requisites  of  a  model  text-book  for  teaching  both  the  science  of 
numbers  and  its  practical  application." — Ibid, 

**The  Intellectual  Arithmetic  is  unquestionably  a  work  of  rare 
excellence."  .  .  .  **  I  am  fuUy  convinced  that  it  has  no  superior 
and  no  equal." — Ibid. 

**The  Primary  Arithmetic  is  happily  adapted  to  lead  the  young 
pupil  to  a  knowledge  of  the  rudiments  of  numbers. — Ibid, 

*' Your  Committee  think  Eaton's  the  best  series  of  Arithmetics  to  be 
had  at  the  present  time." — Boston  Text-Book  Committee,  June,  1864. 

**  Eaton's.Aritlimetics  are  found  to  meet  all  the  wants  of  the  schools, 
and  are  working  well,  —  Ibid,  June,  1865. 

**  I  can  truly  say  that  it  is  the  best  treatise  on  the  subject  I  have  e^'.er 
used." — S.  AV.  Mason,  Master  of  Eliot  School,  Boston, 

**  The  books  deserve  the  wide-spread  popularity  they  have  attained." 
—  H.  II.  Lincoln,  Master  of  Lyman  School,  Boston, 

**It  fully  meets  the  expectations  which  I  entertained  when  it,  was 
introduced." — W.T.  Adams,  Master  of  Boioditch  School,  Boston, 

**I  consider  them  superior  to  any  series  of  the  kind  with  which  I  am 
acquainted." — E.  T.  Quimby,  Professor  of  Mathematics,  Dartmouth 
College, 

"  I  believe  Eaton's  Treatise  (High-School)  far  surpasses  any  other 
work  of  the  kind."  .  ,  .  **  Its  merits  cannot  be  easily  overrated." 
— Albert  C.  Perkins,  Principal  of  High  School,  Lawrence,  Mass, 

** Eaton's  works  give  entire  satisfaction  in  our  schojols."  —  L.  E. 
NoYES,  Superintending  School  Committee,  Abington,  Mass, 

"Eaton's  series  was  introduced  last  October  (1-864)  into  all  the 
schools."    .    .    .     **  The  satisfaction  so  far  is  complete." — J.  Wiii> 
Belcher,  Superintending  Scfiool  Committee,  Randolph,  Mass, 

"The  State  Board  substituted  Eaton's  Common  School  for  Rob';i- 
son's  Practical,  because,  in  thGir  opinion,  it  was  better  adapted  to%he 
wants  of  our  schools."— ^  Hon.  John  Swett,  State  Superintender^of 
Schools,  California. 


4 


**We  think  Eaton  surpasses  any  o-iithor  we  have  ever  used,  and 
believe  his  mathematical  series  is  better  adapted  than  any  other  to  the 
wants  of  our  schools." — John  F.  Colby,  Late  Principal  of  Stetson 
High  School,  Randolph,  Mass, 

**  I  have  never  used  a  work  so  complete  in  all  its  parts  as  the  Com- 
mon School."  .  .  .  *' This  bookstands  high  with  California  teachers." 
— E,.  P.  Foss,  Principal  of  City  Grammar  School,  Sonoi^a,  Col, 

*' Your  Eaton's  series  of  Arithmetics,  so  far  as  examined,  receive 
the  miquahfied  approbation  of  all  teachers." — S.  M.  Shearer,  Prin- 
cipal of  San  Juan  Public  Schools. 

"Eaton's  Arithmetics  are  everywhere  received  with  great  favor." — 
J.  C.  Pelton,  Superintendent  of  Schools,  San  Francisco. 

*  *  After  a  careful  examination  of  all  the  recent  works  on  this  sub- 
ject, they  unanimously  recommended  .  .  .  that  Eaton's  Common- 
School  Arithmetic  be  substituted  for  GreenleaPs.  The  schools  have 
steadily  improved  in  Arithmetic  ever  since  the  change." — Extract 
from  Report  of  Worcester  School  Committee, 

*  *  It  surpasses  all  others  with  which  I  am  acquainted  in  the  following 
particulars."  —  Ephraim  Knight,  Professor  of  Mathematics,  New 
London  Literary  and  Scientific  Institution. 

*  *  Altogether  the  most  satisfactory  that  has  fallen  under  our  notice 
for  practical  use." — Rev.  Lyman  Coleman,  D,D.,  Philadelphia, 

**  The  Board  of  Education  recommended,  at  their  late  meeting,  the 
introduction  of  Eaton's  Arithmetics  into  the  schools  of  New  Hamp- 
shire."— J.  W.  Patterson,  Secretary  of  the  Board  of  Education  for 
New  Hampshire. 

**The  Common-School  Arithmetic  is  just  the  book  for  teaching 
written  Arithmetic  in  all  District  and  Grammar  Schools.  —  J.  D.  Phil- 
brick,  Superintendent  of  Boston  Public  Schools. 

**  The  Intellectual  Arithmetic  is  giving  complete  satisfaction.'^ — C. 
G.  Clarke,  Master  of  Bigelow  School,  Boston. 

**  I  consider  them  the  best  works  on  this  subject  now  extant."  — F. 
F.  Preble,  Snh-Master  Adams  School,  Boston. 

A  valuable  treatise,  well  adapted  to  the  wants  of  public  schools." 
—  J.  N.  Camp,  Superintendent  of  Schools  for  the  State  of  Connecti- 
cut  and  Principal  of  State  Normal  School,  Connecticut. 

*'  The  substitution  of  Eaton  for  Greenleaf  was  voted  without  a 
sin^'-  dissenting  voice." — Rev.  J.  D.  E.  Jones,  Superintendent  of 
Schools,  Worcester,  Mass. 


E  A..  T  o  isr '  s 
STANDARD    SERIES    OF    ARITHMETICS 

IS    RECOMMENDED  IN  HIGH   TERMS   BY   HUNDREDS   OF  THE  BEST   EDUCATORS 
AND  TEACHERS  OF  THE  COUNTRY,  AMONG  WHOM  ARE  THE  FOLLOWING  :.- 

Hon.  John  D.  Philbrick,  Superintendent  of  Public  Schools,  Boston. 

Prof.  J.  P.  Fisk,  Beloit  College,  Beloit,  Wisconsin. 

Hon.  John  Swett,  State  Superintendent  of  Schools  of  California. 

Hon.  James  W.  Patterson,  Secretary  of  N.  H.  State  Board  of  Education. 

William  H.  Wells,  Esq.,  Superintendent  of  Schools,  C|iicago,  Illinois. 

Prof.  A.  Jackman,  Norwich  University,  Norwich,  Vt. 

A.  P.  Stone,  late  President  of  American  Institute  of  Instruction,  Portland,  Me. 

M.  T.  Brown,  Superintendent  of  Schools.  Toledo,  Ohio. 

Hon.  David  N.  Camp,  late  State  Superintendent  of  Schools  of  Connecticut. 

FREDERICK  A.  Sawyer,  Principal  of  High  School,  Charleston,  South  Carolina. 

Prof.  E.  T.  Quimby,  Professor  of  Mathematics,  Dartmouth  College,  N.  H. 

Emory  Lyon,  University  Grammar  School,  Providence,  R.  I. 

Rev.  Charles  Anthon,  D.D.,  Columbia  College,  New  York  City. 

Rev.  Lyman  Coleman,  D.D.,  Seminary,  Philadelphia,  Pa. 

Thomas  Sherwin,  Principal  of  Boston  English   High  School,  and  author  of 

several  works  on  Mathematics. 
Rev.  J.  D.  E.  Jones,  Superintendent  of  Schools,  Worcester,  Mass. 
J.  C.  Pelton,  Superintendent  of  Schools,  San  Francisco,  California. 
J.  W.  EwiNG,  Superintendent  of  Schools,  Perrysburg,  Ohio, 
Prof.  J.  V.  N.  Standish,  Lombard  University,  Galesburg,  111. 

Prof.  S.  F.  Newman,  Principal  of  Normal  School,  Milan,  Ohio. 

J.  E.  Dow,  Superintendent  of  Schools,  Burlington,  Iowa. 

Geo.  W.  Perry,  Superintendent  of  Schools,  Tiffin,  Ohio. 

J.  B.  Roberts,  Superintendent  of  Schools,  Galesburg,  Illinois. 

Rev.  J.  F.  Dudley,  St.  Paul,  Minnesota. 

Albert  C.  Perkins,  Principal  of  High  School,  Lawrence,  Mass. 

Isaac  F.  Cady,  Principal  of  High  School,  Warren,  R.  I. 

Augustus  Morse,  Principal  of  Grammar  School,  Hartford,  Connecticut. 

C.  F.  Emery,  Principal  of  High  School,  Troy,  N.  Y. 

Levi  Cass,  Principal  of  High  School,  Janesville,  Wisconsin. 

Rev.  Arthur  Little,  late  of  North  Haven  Academy,  Connecticut. 

Horace  Day,  Esq.,  late  Superintendent  of  Schools,  New  Haven,  Connecticut. 

Rev.  Walter  S.  Alexander,  Superintendent  of  Schools,  Pomfret,  Connecticut 

John  G.  W.  Martin,  Principal  of  High  School,  Elizabethtown,  Pa. 

C.  P.  Barrows,  Teacher,  Caroline  County,  Va. 

B.  P.  Chenoworth,  Teacher,  Moore's  Hill,  Indiana. 

S.  W.  Mason,  Master  of  Eliot  School,  Boston,  Mass. 

Henry  Freeman,  Commissioner  of  Schools,  Illinois. 


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